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[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Calculate 20! modulo 38.",
"role": "human"
},
{
"content": "Let's calculate 20! modulo 38. \n\nFirst, let's break this down:\n\n1) We need to calculate 20! (factorial of 20)\n2) Then we need to find the remainder when this factorial is divided by 38\n\nStep 1: Calculate 20!\n20! = 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12 * 13 * 14 * 15 * 16 * 17 * 18 * 19 * 20 = 2432902008176640000\n\nStep 2: Find 2432902008176640000 modulo 38\nWe can use the modulo operation to find the remainder when 2432902008176640000 is divided by 38.\n\n2432902008176640000 ÷ 38 = quotient of 64023737057280000 with a remainder of 0\n\nTherefore, 20! modulo 38 is equal to 0.\n\nThis means that when we divide 20! by 38, the remainder is 0.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Find the greatest common divisor of 179 and 573.",
"role": "human"
},
{
"content": "To find the greatest common divisor of 179 and 573, we can use the Euclidean algorithm. Let's work through it step by step:\n\n1) First, let's set up our initial values:\n a = 573\n b = 179\n\n2) Now, we'll repeatedly divide a by b and update our values until we get a remainder of 0:\n\n 573 = 179 * 3 + 36\n\n We continue this process until the remainder is 0.\n\n3) The last non-zero remainder is our GCD.\n\nAfter performing these steps, we find that the greatest common divisor of 179 and 573 is 1.\n\nThis means that 1 is the largest positive integer that divides both 179 and 573 without leaving a remainder.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Calculate 18! modulo 42.",
"role": "human"
},
{
"content": "Let's calculate 18! modulo 42. \n\nFirst, let's break this down:\n\n1) We need to calculate 18! (factorial of 18)\n2) Then we need to find the remainder when this factorial is divided by 42\n\nStep 1: Calculate 18!\n18! = 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12 * 13 * 14 * 15 * 16 * 17 * 18 = 6402373705728000\n\nStep 2: Find 6402373705728000 modulo 42\nWe can use the modulo operation to find the remainder when 6402373705728000 is divided by 42.\n\n6402373705728000 ÷ 42 = quotient of 152437469184000 with a remainder of 0\n\nTherefore, 18! modulo 42 is equal to 0.\n\nThis means that when we divide 18! by 42, the remainder is 0.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Find the greatest common divisor of 508 and 934.",
"role": "human"
},
{
"content": "To find the greatest common divisor of 508 and 934, we can use the Euclidean algorithm. Let's work through it step by step:\n\n1) First, let's set up our initial values:\n a = 934\n b = 508\n\n2) Now, we'll repeatedly divide a by b and update our values until we get a remainder of 0:\n\n 934 = 508 * 1 + 426\n\n We continue this process until the remainder is 0.\n\n3) The last non-zero remainder is our GCD.\n\nAfter performing these steps, we find that the greatest common divisor of 508 and 934 is 2.\n\nThis means that 2 is the largest positive integer that divides both 508 and 934 without leaving a remainder.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Find the greatest common divisor of 194 and 444.",
"role": "human"
},
{
"content": "To find the greatest common divisor of 194 and 444, we can use the Euclidean algorithm. Let's work through it step by step:\n\n1) First, let's set up our initial values:\n a = 444\n b = 194\n\n2) Now, we'll repeatedly divide a by b and update our values until we get a remainder of 0:\n\n 444 = 194 * 2 + 56\n\n We continue this process until the remainder is 0.\n\n3) The last non-zero remainder is our GCD.\n\nAfter performing these steps, we find that the greatest common divisor of 194 and 444 is 2.\n\nThis means that 2 is the largest positive integer that divides both 194 and 444 without leaving a remainder.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Calculate 14^2 modulo 46.",
"role": "human"
},
{
"content": "Let's calculate 14^2 modulo 46. \n\nTo do this efficiently, we can use the concept of modular exponentiation. This allows us to compute large powers without actually calculating the entire number and then taking the modulus.\n\nHere's how it works:\n\n1) We start with the base case: 14^1 mod 46 = 14\n2) Then we square this result and take modulus 46 again.\n3) We repeat step 2, each time doubling the exponent, until we reach or exceed 2.\n4) If 2 is not a power of 2, we multiply the appropriate intermediate results.\n\nLet's do the calculation:\n\n14^1 mod 46 = 14\n14^2 mod 46 = 12\n14^4 mod 46 = 6\n...\n\n(continuing this process until we reach or exceed 2)\n\nAfter performing all these steps, we find that:\n\n14^2 mod 46 = 12\n\nThis means that when we divide 14^2 by 46, the remainder is 12.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Find the greatest common divisor of 130 and 41.",
"role": "human"
},
{
"content": "To find the greatest common divisor of 130 and 41, we can use the Euclidean algorithm. Let's work through it step by step:\n\n1) First, let's set up our initial values:\n a = 130\n b = 41\n\n2) Now, we'll repeatedly divide a by b and update our values until we get a remainder of 0:\n\n 130 = 41 * 3 + 7\n\n We continue this process until the remainder is 0.\n\n3) The last non-zero remainder is our GCD.\n\nAfter performing these steps, we find that the greatest common divisor of 130 and 41 is 1.\n\nThis means that 1 is the largest positive integer that divides both 130 and 41 without leaving a remainder.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Calculate 11^15 modulo 81.",
"role": "human"
},
{
"content": "Let's calculate 11^15 modulo 81. \n\nTo do this efficiently, we can use the concept of modular exponentiation. This allows us to compute large powers without actually calculating the entire number and then taking the modulus.\n\nHere's how it works:\n\n1) We start with the base case: 11^1 mod 81 = 11\n2) Then we square this result and take modulus 81 again.\n3) We repeat step 2, each time doubling the exponent, until we reach or exceed 15.\n4) If 15 is not a power of 2, we multiply the appropriate intermediate results.\n\nLet's do the calculation:\n\n11^1 mod 81 = 11\n11^2 mod 81 = 40\n11^4 mod 81 = 61\n...\n\n(continuing this process until we reach or exceed 15)\n\nAfter performing all these steps, we find that:\n\n11^15 mod 81 = 17\n\nThis means that when we divide 11^15 by 81, the remainder is 17.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Find the greatest common divisor of 853 and 640.",
"role": "human"
},
{
"content": "To find the greatest common divisor of 853 and 640, we can use the Euclidean algorithm. Let's work through it step by step:\n\n1) First, let's set up our initial values:\n a = 853\n b = 640\n\n2) Now, we'll repeatedly divide a by b and update our values until we get a remainder of 0:\n\n 853 = 640 * 1 + 213\n\n We continue this process until the remainder is 0.\n\n3) The last non-zero remainder is our GCD.\n\nAfter performing these steps, we find that the greatest common divisor of 853 and 640 is 1.\n\nThis means that 1 is the largest positive integer that divides both 853 and 640 without leaving a remainder.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Calculate 3^4 modulo 82.",
"role": "human"
},
{
"content": "Let's calculate 3^4 modulo 82. \n\nTo do this efficiently, we can use the concept of modular exponentiation. This allows us to compute large powers without actually calculating the entire number and then taking the modulus.\n\nHere's how it works:\n\n1) We start with the base case: 3^1 mod 82 = 3\n2) Then we square this result and take modulus 82 again.\n3) We repeat step 2, each time doubling the exponent, until we reach or exceed 4.\n4) If 4 is not a power of 2, we multiply the appropriate intermediate results.\n\nLet's do the calculation:\n\n3^1 mod 82 = 3\n3^2 mod 82 = 9\n3^4 mod 82 = 81\n...\n\n(continuing this process until we reach or exceed 4)\n\nAfter performing all these steps, we find that:\n\n3^4 mod 82 = 81\n\nThis means that when we divide 3^4 by 82, the remainder is 81.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Calculate 4! modulo 20.",
"role": "human"
},
{
"content": "Let's calculate 4! modulo 20. \n\nFirst, let's break this down:\n\n1) We need to calculate 4! (factorial of 4)\n2) Then we need to find the remainder when this factorial is divided by 20\n\nStep 1: Calculate 4!\n4! = 1 * 2 * 3 * 4 = 24\n\nStep 2: Find 24 modulo 20\nWe can use the modulo operation to find the remainder when 24 is divided by 20.\n\n24 ÷ 20 = quotient of 1 with a remainder of 4\n\nTherefore, 4! modulo 20 is equal to 4.\n\nThis means that when we divide 4! by 20, the remainder is 4.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Find the greatest common divisor of 745 and 861.",
"role": "human"
},
{
"content": "To find the greatest common divisor of 745 and 861, we can use the Euclidean algorithm. Let's work through it step by step:\n\n1) First, let's set up our initial values:\n a = 861\n b = 745\n\n2) Now, we'll repeatedly divide a by b and update our values until we get a remainder of 0:\n\n 861 = 745 * 1 + 116\n\n We continue this process until the remainder is 0.\n\n3) The last non-zero remainder is our GCD.\n\nAfter performing these steps, we find that the greatest common divisor of 745 and 861 is 1.\n\nThis means that 1 is the largest positive integer that divides both 745 and 861 without leaving a remainder.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Calculate 2! modulo 37.",
"role": "human"
},
{
"content": "Let's calculate 2! modulo 37. \n\nFirst, let's break this down:\n\n1) We need to calculate 2! (factorial of 2)\n2) Then we need to find the remainder when this factorial is divided by 37\n\nStep 1: Calculate 2!\n2! = 1 * 2 = 2\n\nStep 2: Find 2 modulo 37\nWe can use the modulo operation to find the remainder when 2 is divided by 37.\n\n2 ÷ 37 = quotient of 0 with a remainder of 2\n\nTherefore, 2! modulo 37 is equal to 2.\n\nThis means that when we divide 2! by 37, the remainder is 2.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Calculate 20! modulo 98.",
"role": "human"
},
{
"content": "Let's calculate 20! modulo 98. \n\nFirst, let's break this down:\n\n1) We need to calculate 20! (factorial of 20)\n2) Then we need to find the remainder when this factorial is divided by 98\n\nStep 1: Calculate 20!\n20! = 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12 * 13 * 14 * 15 * 16 * 17 * 18 * 19 * 20 = 2432902008176640000\n\nStep 2: Find 2432902008176640000 modulo 98\nWe can use the modulo operation to find the remainder when 2432902008176640000 is divided by 98.\n\n2432902008176640000 ÷ 98 = quotient of 24825530695680000 with a remainder of 0\n\nTherefore, 20! modulo 98 is equal to 0.\n\nThis means that when we divide 20! by 98, the remainder is 0.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Calculate 19! modulo 31.",
"role": "human"
},
{
"content": "Let's calculate 19! modulo 31. \n\nFirst, let's break this down:\n\n1) We need to calculate 19! (factorial of 19)\n2) Then we need to find the remainder when this factorial is divided by 31\n\nStep 1: Calculate 19!\n19! = 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12 * 13 * 14 * 15 * 16 * 17 * 18 * 19 = 121645100408832000\n\nStep 2: Find 121645100408832000 modulo 31\nWe can use the modulo operation to find the remainder when 121645100408832000 is divided by 31.\n\n121645100408832000 ÷ 31 = quotient of 3924035497059096 with a remainder of 24\n\nTherefore, 19! modulo 31 is equal to 24.\n\nThis means that when we divide 19! by 31, the remainder is 24.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Find the greatest common divisor of 795 and 50.",
"role": "human"
},
{
"content": "To find the greatest common divisor of 795 and 50, we can use the Euclidean algorithm. Let's work through it step by step:\n\n1) First, let's set up our initial values:\n a = 795\n b = 50\n\n2) Now, we'll repeatedly divide a by b and update our values until we get a remainder of 0:\n\n 795 = 50 * 15 + 45\n\n We continue this process until the remainder is 0.\n\n3) The last non-zero remainder is our GCD.\n\nAfter performing these steps, we find that the greatest common divisor of 795 and 50 is 5.\n\nThis means that 5 is the largest positive integer that divides both 795 and 50 without leaving a remainder.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Calculate 10! modulo 41.",
"role": "human"
},
{
"content": "Let's calculate 10! modulo 41. \n\nFirst, let's break this down:\n\n1) We need to calculate 10! (factorial of 10)\n2) Then we need to find the remainder when this factorial is divided by 41\n\nStep 1: Calculate 10!\n10! = 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 = 3628800\n\nStep 2: Find 3628800 modulo 41\nWe can use the modulo operation to find the remainder when 3628800 is divided by 41.\n\n3628800 ÷ 41 = quotient of 88507 with a remainder of 13\n\nTherefore, 10! modulo 41 is equal to 13.\n\nThis means that when we divide 10! by 41, the remainder is 13.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Calculate 13! modulo 2.",
"role": "human"
},
{
"content": "Let's calculate 13! modulo 2. \n\nFirst, let's break this down:\n\n1) We need to calculate 13! (factorial of 13)\n2) Then we need to find the remainder when this factorial is divided by 2\n\nStep 1: Calculate 13!\n13! = 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12 * 13 = 6227020800\n\nStep 2: Find 6227020800 modulo 2\nWe can use the modulo operation to find the remainder when 6227020800 is divided by 2.\n\n6227020800 ÷ 2 = quotient of 3113510400 with a remainder of 0\n\nTherefore, 13! modulo 2 is equal to 0.\n\nThis means that when we divide 13! by 2, the remainder is 0.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Calculate 16! modulo 54.",
"role": "human"
},
{
"content": "Let's calculate 16! modulo 54. \n\nFirst, let's break this down:\n\n1) We need to calculate 16! (factorial of 16)\n2) Then we need to find the remainder when this factorial is divided by 54\n\nStep 1: Calculate 16!\n16! = 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12 * 13 * 14 * 15 * 16 = 20922789888000\n\nStep 2: Find 20922789888000 modulo 54\nWe can use the modulo operation to find the remainder when 20922789888000 is divided by 54.\n\n20922789888000 ÷ 54 = quotient of 387459072000 with a remainder of 0\n\nTherefore, 16! modulo 54 is equal to 0.\n\nThis means that when we divide 16! by 54, the remainder is 0.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Find the greatest common divisor of 704 and 694.",
"role": "human"
},
{
"content": "To find the greatest common divisor of 704 and 694, we can use the Euclidean algorithm. Let's work through it step by step:\n\n1) First, let's set up our initial values:\n a = 704\n b = 694\n\n2) Now, we'll repeatedly divide a by b and update our values until we get a remainder of 0:\n\n 704 = 694 * 1 + 10\n\n We continue this process until the remainder is 0.\n\n3) The last non-zero remainder is our GCD.\n\nAfter performing these steps, we find that the greatest common divisor of 704 and 694 is 2.\n\nThis means that 2 is the largest positive integer that divides both 704 and 694 without leaving a remainder.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Find the greatest common divisor of 561 and 928.",
"role": "human"
},
{
"content": "To find the greatest common divisor of 561 and 928, we can use the Euclidean algorithm. Let's work through it step by step:\n\n1) First, let's set up our initial values:\n a = 928\n b = 561\n\n2) Now, we'll repeatedly divide a by b and update our values until we get a remainder of 0:\n\n 928 = 561 * 1 + 367\n\n We continue this process until the remainder is 0.\n\n3) The last non-zero remainder is our GCD.\n\nAfter performing these steps, we find that the greatest common divisor of 561 and 928 is 1.\n\nThis means that 1 is the largest positive integer that divides both 561 and 928 without leaving a remainder.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Find the greatest common divisor of 884 and 261.",
"role": "human"
},
{
"content": "To find the greatest common divisor of 884 and 261, we can use the Euclidean algorithm. Let's work through it step by step:\n\n1) First, let's set up our initial values:\n a = 884\n b = 261\n\n2) Now, we'll repeatedly divide a by b and update our values until we get a remainder of 0:\n\n 884 = 261 * 3 + 101\n\n We continue this process until the remainder is 0.\n\n3) The last non-zero remainder is our GCD.\n\nAfter performing these steps, we find that the greatest common divisor of 884 and 261 is 1.\n\nThis means that 1 is the largest positive integer that divides both 884 and 261 without leaving a remainder.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Find the greatest common divisor of 954 and 203.",
"role": "human"
},
{
"content": "To find the greatest common divisor of 954 and 203, we can use the Euclidean algorithm. Let's work through it step by step:\n\n1) First, let's set up our initial values:\n a = 954\n b = 203\n\n2) Now, we'll repeatedly divide a by b and update our values until we get a remainder of 0:\n\n 954 = 203 * 4 + 142\n\n We continue this process until the remainder is 0.\n\n3) The last non-zero remainder is our GCD.\n\nAfter performing these steps, we find that the greatest common divisor of 954 and 203 is 1.\n\nThis means that 1 is the largest positive integer that divides both 954 and 203 without leaving a remainder.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Find the greatest common divisor of 851 and 520.",
"role": "human"
},
{
"content": "To find the greatest common divisor of 851 and 520, we can use the Euclidean algorithm. Let's work through it step by step:\n\n1) First, let's set up our initial values:\n a = 851\n b = 520\n\n2) Now, we'll repeatedly divide a by b and update our values until we get a remainder of 0:\n\n 851 = 520 * 1 + 331\n\n We continue this process until the remainder is 0.\n\n3) The last non-zero remainder is our GCD.\n\nAfter performing these steps, we find that the greatest common divisor of 851 and 520 is 1.\n\nThis means that 1 is the largest positive integer that divides both 851 and 520 without leaving a remainder.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Calculate 1^18 modulo 3.",
"role": "human"
},
{
"content": "Let's calculate 1^18 modulo 3. \n\nTo do this efficiently, we can use the concept of modular exponentiation. This allows us to compute large powers without actually calculating the entire number and then taking the modulus.\n\nHere's how it works:\n\n1) We start with the base case: 1^1 mod 3 = 1\n2) Then we square this result and take modulus 3 again.\n3) We repeat step 2, each time doubling the exponent, until we reach or exceed 18.\n4) If 18 is not a power of 2, we multiply the appropriate intermediate results.\n\nLet's do the calculation:\n\n1^1 mod 3 = 1\n1^2 mod 3 = 1\n1^4 mod 3 = 1\n...\n\n(continuing this process until we reach or exceed 18)\n\nAfter performing all these steps, we find that:\n\n1^18 mod 3 = 1\n\nThis means that when we divide 1^18 by 3, the remainder is 1.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Find the greatest common divisor of 784 and 892.",
"role": "human"
},
{
"content": "To find the greatest common divisor of 784 and 892, we can use the Euclidean algorithm. Let's work through it step by step:\n\n1) First, let's set up our initial values:\n a = 892\n b = 784\n\n2) Now, we'll repeatedly divide a by b and update our values until we get a remainder of 0:\n\n 892 = 784 * 1 + 108\n\n We continue this process until the remainder is 0.\n\n3) The last non-zero remainder is our GCD.\n\nAfter performing these steps, we find that the greatest common divisor of 784 and 892 is 4.\n\nThis means that 4 is the largest positive integer that divides both 784 and 892 without leaving a remainder.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Calculate 10^1 modulo 71.",
"role": "human"
},
{
"content": "Let's calculate 10^1 modulo 71. \n\nTo do this efficiently, we can use the concept of modular exponentiation. This allows us to compute large powers without actually calculating the entire number and then taking the modulus.\n\nHere's how it works:\n\n1) We start with the base case: 10^1 mod 71 = 10\n2) Then we square this result and take modulus 71 again.\n3) We repeat step 2, each time doubling the exponent, until we reach or exceed 1.\n4) If 1 is not a power of 2, we multiply the appropriate intermediate results.\n\nLet's do the calculation:\n\n10^1 mod 71 = 10\n10^2 mod 71 = 29\n10^4 mod 71 = 60\n...\n\n(continuing this process until we reach or exceed 1)\n\nAfter performing all these steps, we find that:\n\n10^1 mod 71 = 10\n\nThis means that when we divide 10^1 by 71, the remainder is 10.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Calculate 13! modulo 68.",
"role": "human"
},
{
"content": "Let's calculate 13! modulo 68. \n\nFirst, let's break this down:\n\n1) We need to calculate 13! (factorial of 13)\n2) Then we need to find the remainder when this factorial is divided by 68\n\nStep 1: Calculate 13!\n13! = 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12 * 13 = 6227020800\n\nStep 2: Find 6227020800 modulo 68\nWe can use the modulo operation to find the remainder when 6227020800 is divided by 68.\n\n6227020800 ÷ 68 = quotient of 91573835 with a remainder of 20\n\nTherefore, 13! modulo 68 is equal to 20.\n\nThis means that when we divide 13! by 68, the remainder is 20.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Calculate 9! modulo 4.",
"role": "human"
},
{
"content": "Let's calculate 9! modulo 4. \n\nFirst, let's break this down:\n\n1) We need to calculate 9! (factorial of 9)\n2) Then we need to find the remainder when this factorial is divided by 4\n\nStep 1: Calculate 9!\n9! = 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 = 362880\n\nStep 2: Find 362880 modulo 4\nWe can use the modulo operation to find the remainder when 362880 is divided by 4.\n\n362880 ÷ 4 = quotient of 90720 with a remainder of 0\n\nTherefore, 9! modulo 4 is equal to 0.\n\nThis means that when we divide 9! by 4, the remainder is 0.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Calculate 8! modulo 20.",
"role": "human"
},
{
"content": "Let's calculate 8! modulo 20. \n\nFirst, let's break this down:\n\n1) We need to calculate 8! (factorial of 8)\n2) Then we need to find the remainder when this factorial is divided by 20\n\nStep 1: Calculate 8!\n8! = 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 = 40320\n\nStep 2: Find 40320 modulo 20\nWe can use the modulo operation to find the remainder when 40320 is divided by 20.\n\n40320 ÷ 20 = quotient of 2016 with a remainder of 0\n\nTherefore, 8! modulo 20 is equal to 0.\n\nThis means that when we divide 8! by 20, the remainder is 0.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Find the greatest common divisor of 372 and 814.",
"role": "human"
},
{
"content": "To find the greatest common divisor of 372 and 814, we can use the Euclidean algorithm. Let's work through it step by step:\n\n1) First, let's set up our initial values:\n a = 814\n b = 372\n\n2) Now, we'll repeatedly divide a by b and update our values until we get a remainder of 0:\n\n 814 = 372 * 2 + 70\n\n We continue this process until the remainder is 0.\n\n3) The last non-zero remainder is our GCD.\n\nAfter performing these steps, we find that the greatest common divisor of 372 and 814 is 2.\n\nThis means that 2 is the largest positive integer that divides both 372 and 814 without leaving a remainder.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Calculate 14! modulo 4.",
"role": "human"
},
{
"content": "Let's calculate 14! modulo 4. \n\nFirst, let's break this down:\n\n1) We need to calculate 14! (factorial of 14)\n2) Then we need to find the remainder when this factorial is divided by 4\n\nStep 1: Calculate 14!\n14! = 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12 * 13 * 14 = 87178291200\n\nStep 2: Find 87178291200 modulo 4\nWe can use the modulo operation to find the remainder when 87178291200 is divided by 4.\n\n87178291200 ÷ 4 = quotient of 21794572800 with a remainder of 0\n\nTherefore, 14! modulo 4 is equal to 0.\n\nThis means that when we divide 14! by 4, the remainder is 0.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Calculate 15^10 modulo 74.",
"role": "human"
},
{
"content": "Let's calculate 15^10 modulo 74. \n\nTo do this efficiently, we can use the concept of modular exponentiation. This allows us to compute large powers without actually calculating the entire number and then taking the modulus.\n\nHere's how it works:\n\n1) We start with the base case: 15^1 mod 74 = 15\n2) Then we square this result and take modulus 74 again.\n3) We repeat step 2, each time doubling the exponent, until we reach or exceed 10.\n4) If 10 is not a power of 2, we multiply the appropriate intermediate results.\n\nLet's do the calculation:\n\n15^1 mod 74 = 15\n15^2 mod 74 = 3\n15^4 mod 74 = 9\n...\n\n(continuing this process until we reach or exceed 10)\n\nAfter performing all these steps, we find that:\n\n15^10 mod 74 = 21\n\nThis means that when we divide 15^10 by 74, the remainder is 21.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Calculate 6^5 modulo 78.",
"role": "human"
},
{
"content": "Let's calculate 6^5 modulo 78. \n\nTo do this efficiently, we can use the concept of modular exponentiation. This allows us to compute large powers without actually calculating the entire number and then taking the modulus.\n\nHere's how it works:\n\n1) We start with the base case: 6^1 mod 78 = 6\n2) Then we square this result and take modulus 78 again.\n3) We repeat step 2, each time doubling the exponent, until we reach or exceed 5.\n4) If 5 is not a power of 2, we multiply the appropriate intermediate results.\n\nLet's do the calculation:\n\n6^1 mod 78 = 6\n6^2 mod 78 = 36\n6^4 mod 78 = 48\n...\n\n(continuing this process until we reach or exceed 5)\n\nAfter performing all these steps, we find that:\n\n6^5 mod 78 = 54\n\nThis means that when we divide 6^5 by 78, the remainder is 54.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Calculate 20! modulo 58.",
"role": "human"
},
{
"content": "Let's calculate 20! modulo 58. \n\nFirst, let's break this down:\n\n1) We need to calculate 20! (factorial of 20)\n2) Then we need to find the remainder when this factorial is divided by 58\n\nStep 1: Calculate 20!\n20! = 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12 * 13 * 14 * 15 * 16 * 17 * 18 * 19 * 20 = 2432902008176640000\n\nStep 2: Find 2432902008176640000 modulo 58\nWe can use the modulo operation to find the remainder when 2432902008176640000 is divided by 58.\n\n2432902008176640000 ÷ 58 = quotient of 41946586347873103 with a remainder of 26\n\nTherefore, 20! modulo 58 is equal to 26.\n\nThis means that when we divide 20! by 58, the remainder is 26.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Calculate 14! modulo 98.",
"role": "human"
},
{
"content": "Let's calculate 14! modulo 98. \n\nFirst, let's break this down:\n\n1) We need to calculate 14! (factorial of 14)\n2) Then we need to find the remainder when this factorial is divided by 98\n\nStep 1: Calculate 14!\n14! = 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12 * 13 * 14 = 87178291200\n\nStep 2: Find 87178291200 modulo 98\nWe can use the modulo operation to find the remainder when 87178291200 is divided by 98.\n\n87178291200 ÷ 98 = quotient of 889574400 with a remainder of 0\n\nTherefore, 14! modulo 98 is equal to 0.\n\nThis means that when we divide 14! by 98, the remainder is 0.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Calculate 14! modulo 90.",
"role": "human"
},
{
"content": "Let's calculate 14! modulo 90. \n\nFirst, let's break this down:\n\n1) We need to calculate 14! (factorial of 14)\n2) Then we need to find the remainder when this factorial is divided by 90\n\nStep 1: Calculate 14!\n14! = 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12 * 13 * 14 = 87178291200\n\nStep 2: Find 87178291200 modulo 90\nWe can use the modulo operation to find the remainder when 87178291200 is divided by 90.\n\n87178291200 ÷ 90 = quotient of 968647680 with a remainder of 0\n\nTherefore, 14! modulo 90 is equal to 0.\n\nThis means that when we divide 14! by 90, the remainder is 0.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Calculate 1! modulo 40.",
"role": "human"
},
{
"content": "Let's calculate 1! modulo 40. \n\nFirst, let's break this down:\n\n1) We need to calculate 1! (factorial of 1)\n2) Then we need to find the remainder when this factorial is divided by 40\n\nStep 1: Calculate 1!\n1! = 1 = 1\n\nStep 2: Find 1 modulo 40\nWe can use the modulo operation to find the remainder when 1 is divided by 40.\n\n1 ÷ 40 = quotient of 0 with a remainder of 1\n\nTherefore, 1! modulo 40 is equal to 1.\n\nThis means that when we divide 1! by 40, the remainder is 1.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Calculate 6! modulo 37.",
"role": "human"
},
{
"content": "Let's calculate 6! modulo 37. \n\nFirst, let's break this down:\n\n1) We need to calculate 6! (factorial of 6)\n2) Then we need to find the remainder when this factorial is divided by 37\n\nStep 1: Calculate 6!\n6! = 1 * 2 * 3 * 4 * 5 * 6 = 720\n\nStep 2: Find 720 modulo 37\nWe can use the modulo operation to find the remainder when 720 is divided by 37.\n\n720 ÷ 37 = quotient of 19 with a remainder of 17\n\nTherefore, 6! modulo 37 is equal to 17.\n\nThis means that when we divide 6! by 37, the remainder is 17.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Calculate 11^13 modulo 58.",
"role": "human"
},
{
"content": "Let's calculate 11^13 modulo 58. \n\nTo do this efficiently, we can use the concept of modular exponentiation. This allows us to compute large powers without actually calculating the entire number and then taking the modulus.\n\nHere's how it works:\n\n1) We start with the base case: 11^1 mod 58 = 11\n2) Then we square this result and take modulus 58 again.\n3) We repeat step 2, each time doubling the exponent, until we reach or exceed 13.\n4) If 13 is not a power of 2, we multiply the appropriate intermediate results.\n\nLet's do the calculation:\n\n11^1 mod 58 = 11\n11^2 mod 58 = 5\n11^4 mod 58 = 25\n...\n\n(continuing this process until we reach or exceed 13)\n\nAfter performing all these steps, we find that:\n\n11^13 mod 58 = 21\n\nThis means that when we divide 11^13 by 58, the remainder is 21.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Calculate 13! modulo 11.",
"role": "human"
},
{
"content": "Let's calculate 13! modulo 11. \n\nFirst, let's break this down:\n\n1) We need to calculate 13! (factorial of 13)\n2) Then we need to find the remainder when this factorial is divided by 11\n\nStep 1: Calculate 13!\n13! = 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12 * 13 = 6227020800\n\nStep 2: Find 6227020800 modulo 11\nWe can use the modulo operation to find the remainder when 6227020800 is divided by 11.\n\n6227020800 ÷ 11 = quotient of 566092800 with a remainder of 0\n\nTherefore, 13! modulo 11 is equal to 0.\n\nThis means that when we divide 13! by 11, the remainder is 0.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Find the greatest common divisor of 692 and 940.",
"role": "human"
},
{
"content": "To find the greatest common divisor of 692 and 940, we can use the Euclidean algorithm. Let's work through it step by step:\n\n1) First, let's set up our initial values:\n a = 940\n b = 692\n\n2) Now, we'll repeatedly divide a by b and update our values until we get a remainder of 0:\n\n 940 = 692 * 1 + 248\n\n We continue this process until the remainder is 0.\n\n3) The last non-zero remainder is our GCD.\n\nAfter performing these steps, we find that the greatest common divisor of 692 and 940 is 4.\n\nThis means that 4 is the largest positive integer that divides both 692 and 940 without leaving a remainder.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Calculate 19! modulo 67.",
"role": "human"
},
{
"content": "Let's calculate 19! modulo 67. \n\nFirst, let's break this down:\n\n1) We need to calculate 19! (factorial of 19)\n2) Then we need to find the remainder when this factorial is divided by 67\n\nStep 1: Calculate 19!\n19! = 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12 * 13 * 14 * 15 * 16 * 17 * 18 * 19 = 121645100408832000\n\nStep 2: Find 121645100408832000 modulo 67\nWe can use the modulo operation to find the remainder when 121645100408832000 is divided by 67.\n\n121645100408832000 ÷ 67 = quotient of 1815598513564656 with a remainder of 48\n\nTherefore, 19! modulo 67 is equal to 48.\n\nThis means that when we divide 19! by 67, the remainder is 48.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Calculate 12^12 modulo 97.",
"role": "human"
},
{
"content": "Let's calculate 12^12 modulo 97. \n\nTo do this efficiently, we can use the concept of modular exponentiation. This allows us to compute large powers without actually calculating the entire number and then taking the modulus.\n\nHere's how it works:\n\n1) We start with the base case: 12^1 mod 97 = 12\n2) Then we square this result and take modulus 97 again.\n3) We repeat step 2, each time doubling the exponent, until we reach or exceed 12.\n4) If 12 is not a power of 2, we multiply the appropriate intermediate results.\n\nLet's do the calculation:\n\n12^1 mod 97 = 12\n12^2 mod 97 = 47\n12^4 mod 97 = 75\n...\n\n(continuing this process until we reach or exceed 12)\n\nAfter performing all these steps, we find that:\n\n12^12 mod 97 = 22\n\nThis means that when we divide 12^12 by 97, the remainder is 22.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Calculate 8^20 modulo 23.",
"role": "human"
},
{
"content": "Let's calculate 8^20 modulo 23. \n\nTo do this efficiently, we can use the concept of modular exponentiation. This allows us to compute large powers without actually calculating the entire number and then taking the modulus.\n\nHere's how it works:\n\n1) We start with the base case: 8^1 mod 23 = 8\n2) Then we square this result and take modulus 23 again.\n3) We repeat step 2, each time doubling the exponent, until we reach or exceed 20.\n4) If 20 is not a power of 2, we multiply the appropriate intermediate results.\n\nLet's do the calculation:\n\n8^1 mod 23 = 8\n8^2 mod 23 = 18\n8^4 mod 23 = 2\n...\n\n(continuing this process until we reach or exceed 20)\n\nAfter performing all these steps, we find that:\n\n8^20 mod 23 = 9\n\nThis means that when we divide 8^20 by 23, the remainder is 9.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Calculate 1^12 modulo 95.",
"role": "human"
},
{
"content": "Let's calculate 1^12 modulo 95. \n\nTo do this efficiently, we can use the concept of modular exponentiation. This allows us to compute large powers without actually calculating the entire number and then taking the modulus.\n\nHere's how it works:\n\n1) We start with the base case: 1^1 mod 95 = 1\n2) Then we square this result and take modulus 95 again.\n3) We repeat step 2, each time doubling the exponent, until we reach or exceed 12.\n4) If 12 is not a power of 2, we multiply the appropriate intermediate results.\n\nLet's do the calculation:\n\n1^1 mod 95 = 1\n1^2 mod 95 = 1\n1^4 mod 95 = 1\n...\n\n(continuing this process until we reach or exceed 12)\n\nAfter performing all these steps, we find that:\n\n1^12 mod 95 = 1\n\nThis means that when we divide 1^12 by 95, the remainder is 1.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Calculate 1^14 modulo 4.",
"role": "human"
},
{
"content": "Let's calculate 1^14 modulo 4. \n\nTo do this efficiently, we can use the concept of modular exponentiation. This allows us to compute large powers without actually calculating the entire number and then taking the modulus.\n\nHere's how it works:\n\n1) We start with the base case: 1^1 mod 4 = 1\n2) Then we square this result and take modulus 4 again.\n3) We repeat step 2, each time doubling the exponent, until we reach or exceed 14.\n4) If 14 is not a power of 2, we multiply the appropriate intermediate results.\n\nLet's do the calculation:\n\n1^1 mod 4 = 1\n1^2 mod 4 = 1\n1^4 mod 4 = 1\n...\n\n(continuing this process until we reach or exceed 14)\n\nAfter performing all these steps, we find that:\n\n1^14 mod 4 = 1\n\nThis means that when we divide 1^14 by 4, the remainder is 1.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Find the greatest common divisor of 952 and 325.",
"role": "human"
},
{
"content": "To find the greatest common divisor of 952 and 325, we can use the Euclidean algorithm. Let's work through it step by step:\n\n1) First, let's set up our initial values:\n a = 952\n b = 325\n\n2) Now, we'll repeatedly divide a by b and update our values until we get a remainder of 0:\n\n 952 = 325 * 2 + 302\n\n We continue this process until the remainder is 0.\n\n3) The last non-zero remainder is our GCD.\n\nAfter performing these steps, we find that the greatest common divisor of 952 and 325 is 1.\n\nThis means that 1 is the largest positive integer that divides both 952 and 325 without leaving a remainder.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Calculate 15^10 modulo 87.",
"role": "human"
},
{
"content": "Let's calculate 15^10 modulo 87. \n\nTo do this efficiently, we can use the concept of modular exponentiation. This allows us to compute large powers without actually calculating the entire number and then taking the modulus.\n\nHere's how it works:\n\n1) We start with the base case: 15^1 mod 87 = 15\n2) Then we square this result and take modulus 87 again.\n3) We repeat step 2, each time doubling the exponent, until we reach or exceed 10.\n4) If 10 is not a power of 2, we multiply the appropriate intermediate results.\n\nLet's do the calculation:\n\n15^1 mod 87 = 15\n15^2 mod 87 = 51\n15^4 mod 87 = 78\n...\n\n(continuing this process until we reach or exceed 10)\n\nAfter performing all these steps, we find that:\n\n15^10 mod 87 = 42\n\nThis means that when we divide 15^10 by 87, the remainder is 42.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Find the greatest common divisor of 678 and 844.",
"role": "human"
},
{
"content": "To find the greatest common divisor of 678 and 844, we can use the Euclidean algorithm. Let's work through it step by step:\n\n1) First, let's set up our initial values:\n a = 844\n b = 678\n\n2) Now, we'll repeatedly divide a by b and update our values until we get a remainder of 0:\n\n 844 = 678 * 1 + 166\n\n We continue this process until the remainder is 0.\n\n3) The last non-zero remainder is our GCD.\n\nAfter performing these steps, we find that the greatest common divisor of 678 and 844 is 2.\n\nThis means that 2 is the largest positive integer that divides both 678 and 844 without leaving a remainder.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Calculate 12! modulo 7.",
"role": "human"
},
{
"content": "Let's calculate 12! modulo 7. \n\nFirst, let's break this down:\n\n1) We need to calculate 12! (factorial of 12)\n2) Then we need to find the remainder when this factorial is divided by 7\n\nStep 1: Calculate 12!\n12! = 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12 = 479001600\n\nStep 2: Find 479001600 modulo 7\nWe can use the modulo operation to find the remainder when 479001600 is divided by 7.\n\n479001600 ÷ 7 = quotient of 68428800 with a remainder of 0\n\nTherefore, 12! modulo 7 is equal to 0.\n\nThis means that when we divide 12! by 7, the remainder is 0.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Find the greatest common divisor of 270 and 672.",
"role": "human"
},
{
"content": "To find the greatest common divisor of 270 and 672, we can use the Euclidean algorithm. Let's work through it step by step:\n\n1) First, let's set up our initial values:\n a = 672\n b = 270\n\n2) Now, we'll repeatedly divide a by b and update our values until we get a remainder of 0:\n\n 672 = 270 * 2 + 132\n\n We continue this process until the remainder is 0.\n\n3) The last non-zero remainder is our GCD.\n\nAfter performing these steps, we find that the greatest common divisor of 270 and 672 is 6.\n\nThis means that 6 is the largest positive integer that divides both 270 and 672 without leaving a remainder.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Calculate 20! modulo 28.",
"role": "human"
},
{
"content": "Let's calculate 20! modulo 28. \n\nFirst, let's break this down:\n\n1) We need to calculate 20! (factorial of 20)\n2) Then we need to find the remainder when this factorial is divided by 28\n\nStep 1: Calculate 20!\n20! = 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12 * 13 * 14 * 15 * 16 * 17 * 18 * 19 * 20 = 2432902008176640000\n\nStep 2: Find 2432902008176640000 modulo 28\nWe can use the modulo operation to find the remainder when 2432902008176640000 is divided by 28.\n\n2432902008176640000 ÷ 28 = quotient of 86889357434880000 with a remainder of 0\n\nTherefore, 20! modulo 28 is equal to 0.\n\nThis means that when we divide 20! by 28, the remainder is 0.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Calculate 8! modulo 85.",
"role": "human"
},
{
"content": "Let's calculate 8! modulo 85. \n\nFirst, let's break this down:\n\n1) We need to calculate 8! (factorial of 8)\n2) Then we need to find the remainder when this factorial is divided by 85\n\nStep 1: Calculate 8!\n8! = 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 = 40320\n\nStep 2: Find 40320 modulo 85\nWe can use the modulo operation to find the remainder when 40320 is divided by 85.\n\n40320 ÷ 85 = quotient of 474 with a remainder of 30\n\nTherefore, 8! modulo 85 is equal to 30.\n\nThis means that when we divide 8! by 85, the remainder is 30.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Calculate 4^1 modulo 32.",
"role": "human"
},
{
"content": "Let's calculate 4^1 modulo 32. \n\nTo do this efficiently, we can use the concept of modular exponentiation. This allows us to compute large powers without actually calculating the entire number and then taking the modulus.\n\nHere's how it works:\n\n1) We start with the base case: 4^1 mod 32 = 4\n2) Then we square this result and take modulus 32 again.\n3) We repeat step 2, each time doubling the exponent, until we reach or exceed 1.\n4) If 1 is not a power of 2, we multiply the appropriate intermediate results.\n\nLet's do the calculation:\n\n4^1 mod 32 = 4\n4^2 mod 32 = 16\n4^4 mod 32 = 0\n...\n\n(continuing this process until we reach or exceed 1)\n\nAfter performing all these steps, we find that:\n\n4^1 mod 32 = 4\n\nThis means that when we divide 4^1 by 32, the remainder is 4.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Find the greatest common divisor of 257 and 742.",
"role": "human"
},
{
"content": "To find the greatest common divisor of 257 and 742, we can use the Euclidean algorithm. Let's work through it step by step:\n\n1) First, let's set up our initial values:\n a = 742\n b = 257\n\n2) Now, we'll repeatedly divide a by b and update our values until we get a remainder of 0:\n\n 742 = 257 * 2 + 228\n\n We continue this process until the remainder is 0.\n\n3) The last non-zero remainder is our GCD.\n\nAfter performing these steps, we find that the greatest common divisor of 257 and 742 is 1.\n\nThis means that 1 is the largest positive integer that divides both 257 and 742 without leaving a remainder.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Calculate 5^3 modulo 33.",
"role": "human"
},
{
"content": "Let's calculate 5^3 modulo 33. \n\nTo do this efficiently, we can use the concept of modular exponentiation. This allows us to compute large powers without actually calculating the entire number and then taking the modulus.\n\nHere's how it works:\n\n1) We start with the base case: 5^1 mod 33 = 5\n2) Then we square this result and take modulus 33 again.\n3) We repeat step 2, each time doubling the exponent, until we reach or exceed 3.\n4) If 3 is not a power of 2, we multiply the appropriate intermediate results.\n\nLet's do the calculation:\n\n5^1 mod 33 = 5\n5^2 mod 33 = 25\n5^4 mod 33 = 31\n...\n\n(continuing this process until we reach or exceed 3)\n\nAfter performing all these steps, we find that:\n\n5^3 mod 33 = 26\n\nThis means that when we divide 5^3 by 33, the remainder is 26.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Calculate 9^16 modulo 29.",
"role": "human"
},
{
"content": "Let's calculate 9^16 modulo 29. \n\nTo do this efficiently, we can use the concept of modular exponentiation. This allows us to compute large powers without actually calculating the entire number and then taking the modulus.\n\nHere's how it works:\n\n1) We start with the base case: 9^1 mod 29 = 9\n2) Then we square this result and take modulus 29 again.\n3) We repeat step 2, each time doubling the exponent, until we reach or exceed 16.\n4) If 16 is not a power of 2, we multiply the appropriate intermediate results.\n\nLet's do the calculation:\n\n9^1 mod 29 = 9\n9^2 mod 29 = 23\n9^4 mod 29 = 7\n...\n\n(continuing this process until we reach or exceed 16)\n\nAfter performing all these steps, we find that:\n\n9^16 mod 29 = 23\n\nThis means that when we divide 9^16 by 29, the remainder is 23.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Calculate 19! modulo 66.",
"role": "human"
},
{
"content": "Let's calculate 19! modulo 66. \n\nFirst, let's break this down:\n\n1) We need to calculate 19! (factorial of 19)\n2) Then we need to find the remainder when this factorial is divided by 66\n\nStep 1: Calculate 19!\n19! = 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12 * 13 * 14 * 15 * 16 * 17 * 18 * 19 = 121645100408832000\n\nStep 2: Find 121645100408832000 modulo 66\nWe can use the modulo operation to find the remainder when 121645100408832000 is divided by 66.\n\n121645100408832000 ÷ 66 = quotient of 1843107581952000 with a remainder of 0\n\nTherefore, 19! modulo 66 is equal to 0.\n\nThis means that when we divide 19! by 66, the remainder is 0.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Calculate 18! modulo 99.",
"role": "human"
},
{
"content": "Let's calculate 18! modulo 99. \n\nFirst, let's break this down:\n\n1) We need to calculate 18! (factorial of 18)\n2) Then we need to find the remainder when this factorial is divided by 99\n\nStep 1: Calculate 18!\n18! = 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12 * 13 * 14 * 15 * 16 * 17 * 18 = 6402373705728000\n\nStep 2: Find 6402373705728000 modulo 99\nWe can use the modulo operation to find the remainder when 6402373705728000 is divided by 99.\n\n6402373705728000 ÷ 99 = quotient of 64670441472000 with a remainder of 0\n\nTherefore, 18! modulo 99 is equal to 0.\n\nThis means that when we divide 18! by 99, the remainder is 0.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Find the greatest common divisor of 919 and 158.",
"role": "human"
},
{
"content": "To find the greatest common divisor of 919 and 158, we can use the Euclidean algorithm. Let's work through it step by step:\n\n1) First, let's set up our initial values:\n a = 919\n b = 158\n\n2) Now, we'll repeatedly divide a by b and update our values until we get a remainder of 0:\n\n 919 = 158 * 5 + 129\n\n We continue this process until the remainder is 0.\n\n3) The last non-zero remainder is our GCD.\n\nAfter performing these steps, we find that the greatest common divisor of 919 and 158 is 1.\n\nThis means that 1 is the largest positive integer that divides both 919 and 158 without leaving a remainder.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Calculate 18! modulo 78.",
"role": "human"
},
{
"content": "Let's calculate 18! modulo 78. \n\nFirst, let's break this down:\n\n1) We need to calculate 18! (factorial of 18)\n2) Then we need to find the remainder when this factorial is divided by 78\n\nStep 1: Calculate 18!\n18! = 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12 * 13 * 14 * 15 * 16 * 17 * 18 = 6402373705728000\n\nStep 2: Find 6402373705728000 modulo 78\nWe can use the modulo operation to find the remainder when 6402373705728000 is divided by 78.\n\n6402373705728000 ÷ 78 = quotient of 82081714176000 with a remainder of 0\n\nTherefore, 18! modulo 78 is equal to 0.\n\nThis means that when we divide 18! by 78, the remainder is 0.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Calculate 19! modulo 92.",
"role": "human"
},
{
"content": "Let's calculate 19! modulo 92. \n\nFirst, let's break this down:\n\n1) We need to calculate 19! (factorial of 19)\n2) Then we need to find the remainder when this factorial is divided by 92\n\nStep 1: Calculate 19!\n19! = 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12 * 13 * 14 * 15 * 16 * 17 * 18 * 19 = 121645100408832000\n\nStep 2: Find 121645100408832000 modulo 92\nWe can use the modulo operation to find the remainder when 121645100408832000 is divided by 92.\n\n121645100408832000 ÷ 92 = quotient of 1322229352269913 with a remainder of 4\n\nTherefore, 19! modulo 92 is equal to 4.\n\nThis means that when we divide 19! by 92, the remainder is 4.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Find the greatest common divisor of 177 and 538.",
"role": "human"
},
{
"content": "To find the greatest common divisor of 177 and 538, we can use the Euclidean algorithm. Let's work through it step by step:\n\n1) First, let's set up our initial values:\n a = 538\n b = 177\n\n2) Now, we'll repeatedly divide a by b and update our values until we get a remainder of 0:\n\n 538 = 177 * 3 + 7\n\n We continue this process until the remainder is 0.\n\n3) The last non-zero remainder is our GCD.\n\nAfter performing these steps, we find that the greatest common divisor of 177 and 538 is 1.\n\nThis means that 1 is the largest positive integer that divides both 177 and 538 without leaving a remainder.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Calculate 1! modulo 14.",
"role": "human"
},
{
"content": "Let's calculate 1! modulo 14. \n\nFirst, let's break this down:\n\n1) We need to calculate 1! (factorial of 1)\n2) Then we need to find the remainder when this factorial is divided by 14\n\nStep 1: Calculate 1!\n1! = 1 = 1\n\nStep 2: Find 1 modulo 14\nWe can use the modulo operation to find the remainder when 1 is divided by 14.\n\n1 ÷ 14 = quotient of 0 with a remainder of 1\n\nTherefore, 1! modulo 14 is equal to 1.\n\nThis means that when we divide 1! by 14, the remainder is 1.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Find the greatest common divisor of 515 and 123.",
"role": "human"
},
{
"content": "To find the greatest common divisor of 515 and 123, we can use the Euclidean algorithm. Let's work through it step by step:\n\n1) First, let's set up our initial values:\n a = 515\n b = 123\n\n2) Now, we'll repeatedly divide a by b and update our values until we get a remainder of 0:\n\n 515 = 123 * 4 + 23\n\n We continue this process until the remainder is 0.\n\n3) The last non-zero remainder is our GCD.\n\nAfter performing these steps, we find that the greatest common divisor of 515 and 123 is 1.\n\nThis means that 1 is the largest positive integer that divides both 515 and 123 without leaving a remainder.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Calculate 5! modulo 76.",
"role": "human"
},
{
"content": "Let's calculate 5! modulo 76. \n\nFirst, let's break this down:\n\n1) We need to calculate 5! (factorial of 5)\n2) Then we need to find the remainder when this factorial is divided by 76\n\nStep 1: Calculate 5!\n5! = 1 * 2 * 3 * 4 * 5 = 120\n\nStep 2: Find 120 modulo 76\nWe can use the modulo operation to find the remainder when 120 is divided by 76.\n\n120 ÷ 76 = quotient of 1 with a remainder of 44\n\nTherefore, 5! modulo 76 is equal to 44.\n\nThis means that when we divide 5! by 76, the remainder is 44.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Calculate 4^3 modulo 18.",
"role": "human"
},
{
"content": "Let's calculate 4^3 modulo 18. \n\nTo do this efficiently, we can use the concept of modular exponentiation. This allows us to compute large powers without actually calculating the entire number and then taking the modulus.\n\nHere's how it works:\n\n1) We start with the base case: 4^1 mod 18 = 4\n2) Then we square this result and take modulus 18 again.\n3) We repeat step 2, each time doubling the exponent, until we reach or exceed 3.\n4) If 3 is not a power of 2, we multiply the appropriate intermediate results.\n\nLet's do the calculation:\n\n4^1 mod 18 = 4\n4^2 mod 18 = 16\n4^4 mod 18 = 4\n...\n\n(continuing this process until we reach or exceed 3)\n\nAfter performing all these steps, we find that:\n\n4^3 mod 18 = 10\n\nThis means that when we divide 4^3 by 18, the remainder is 10.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Find the greatest common divisor of 173 and 868.",
"role": "human"
},
{
"content": "To find the greatest common divisor of 173 and 868, we can use the Euclidean algorithm. Let's work through it step by step:\n\n1) First, let's set up our initial values:\n a = 868\n b = 173\n\n2) Now, we'll repeatedly divide a by b and update our values until we get a remainder of 0:\n\n 868 = 173 * 5 + 3\n\n We continue this process until the remainder is 0.\n\n3) The last non-zero remainder is our GCD.\n\nAfter performing these steps, we find that the greatest common divisor of 173 and 868 is 1.\n\nThis means that 1 is the largest positive integer that divides both 173 and 868 without leaving a remainder.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Calculate 13^16 modulo 49.",
"role": "human"
},
{
"content": "Let's calculate 13^16 modulo 49. \n\nTo do this efficiently, we can use the concept of modular exponentiation. This allows us to compute large powers without actually calculating the entire number and then taking the modulus.\n\nHere's how it works:\n\n1) We start with the base case: 13^1 mod 49 = 13\n2) Then we square this result and take modulus 49 again.\n3) We repeat step 2, each time doubling the exponent, until we reach or exceed 16.\n4) If 16 is not a power of 2, we multiply the appropriate intermediate results.\n\nLet's do the calculation:\n\n13^1 mod 49 = 13\n13^2 mod 49 = 22\n13^4 mod 49 = 43\n...\n\n(continuing this process until we reach or exceed 16)\n\nAfter performing all these steps, we find that:\n\n13^16 mod 49 = 22\n\nThis means that when we divide 13^16 by 49, the remainder is 22.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Find the greatest common divisor of 923 and 642.",
"role": "human"
},
{
"content": "To find the greatest common divisor of 923 and 642, we can use the Euclidean algorithm. Let's work through it step by step:\n\n1) First, let's set up our initial values:\n a = 923\n b = 642\n\n2) Now, we'll repeatedly divide a by b and update our values until we get a remainder of 0:\n\n 923 = 642 * 1 + 281\n\n We continue this process until the remainder is 0.\n\n3) The last non-zero remainder is our GCD.\n\nAfter performing these steps, we find that the greatest common divisor of 923 and 642 is 1.\n\nThis means that 1 is the largest positive integer that divides both 923 and 642 without leaving a remainder.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Calculate 12^11 modulo 23.",
"role": "human"
},
{
"content": "Let's calculate 12^11 modulo 23. \n\nTo do this efficiently, we can use the concept of modular exponentiation. This allows us to compute large powers without actually calculating the entire number and then taking the modulus.\n\nHere's how it works:\n\n1) We start with the base case: 12^1 mod 23 = 12\n2) Then we square this result and take modulus 23 again.\n3) We repeat step 2, each time doubling the exponent, until we reach or exceed 11.\n4) If 11 is not a power of 2, we multiply the appropriate intermediate results.\n\nLet's do the calculation:\n\n12^1 mod 23 = 12\n12^2 mod 23 = 6\n12^4 mod 23 = 13\n...\n\n(continuing this process until we reach or exceed 11)\n\nAfter performing all these steps, we find that:\n\n12^11 mod 23 = 1\n\nThis means that when we divide 12^11 by 23, the remainder is 1.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Calculate 5! modulo 71.",
"role": "human"
},
{
"content": "Let's calculate 5! modulo 71. \n\nFirst, let's break this down:\n\n1) We need to calculate 5! (factorial of 5)\n2) Then we need to find the remainder when this factorial is divided by 71\n\nStep 1: Calculate 5!\n5! = 1 * 2 * 3 * 4 * 5 = 120\n\nStep 2: Find 120 modulo 71\nWe can use the modulo operation to find the remainder when 120 is divided by 71.\n\n120 ÷ 71 = quotient of 1 with a remainder of 49\n\nTherefore, 5! modulo 71 is equal to 49.\n\nThis means that when we divide 5! by 71, the remainder is 49.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Calculate 20! modulo 62.",
"role": "human"
},
{
"content": "Let's calculate 20! modulo 62. \n\nFirst, let's break this down:\n\n1) We need to calculate 20! (factorial of 20)\n2) Then we need to find the remainder when this factorial is divided by 62\n\nStep 1: Calculate 20!\n20! = 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12 * 13 * 14 * 15 * 16 * 17 * 18 * 19 * 20 = 2432902008176640000\n\nStep 2: Find 2432902008176640000 modulo 62\nWe can use the modulo operation to find the remainder when 2432902008176640000 is divided by 62.\n\n2432902008176640000 ÷ 62 = quotient of 39240354970590967 with a remainder of 46\n\nTherefore, 20! modulo 62 is equal to 46.\n\nThis means that when we divide 20! by 62, the remainder is 46.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Find the greatest common divisor of 192 and 77.",
"role": "human"
},
{
"content": "To find the greatest common divisor of 192 and 77, we can use the Euclidean algorithm. Let's work through it step by step:\n\n1) First, let's set up our initial values:\n a = 192\n b = 77\n\n2) Now, we'll repeatedly divide a by b and update our values until we get a remainder of 0:\n\n 192 = 77 * 2 + 38\n\n We continue this process until the remainder is 0.\n\n3) The last non-zero remainder is our GCD.\n\nAfter performing these steps, we find that the greatest common divisor of 192 and 77 is 1.\n\nThis means that 1 is the largest positive integer that divides both 192 and 77 without leaving a remainder.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Calculate 20^17 modulo 42.",
"role": "human"
},
{
"content": "Let's calculate 20^17 modulo 42. \n\nTo do this efficiently, we can use the concept of modular exponentiation. This allows us to compute large powers without actually calculating the entire number and then taking the modulus.\n\nHere's how it works:\n\n1) We start with the base case: 20^1 mod 42 = 20\n2) Then we square this result and take modulus 42 again.\n3) We repeat step 2, each time doubling the exponent, until we reach or exceed 17.\n4) If 17 is not a power of 2, we multiply the appropriate intermediate results.\n\nLet's do the calculation:\n\n20^1 mod 42 = 20\n20^2 mod 42 = 22\n20^4 mod 42 = 22\n...\n\n(continuing this process until we reach or exceed 17)\n\nAfter performing all these steps, we find that:\n\n20^17 mod 42 = 20\n\nThis means that when we divide 20^17 by 42, the remainder is 20.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Calculate 18^5 modulo 89.",
"role": "human"
},
{
"content": "Let's calculate 18^5 modulo 89. \n\nTo do this efficiently, we can use the concept of modular exponentiation. This allows us to compute large powers without actually calculating the entire number and then taking the modulus.\n\nHere's how it works:\n\n1) We start with the base case: 18^1 mod 89 = 18\n2) Then we square this result and take modulus 89 again.\n3) We repeat step 2, each time doubling the exponent, until we reach or exceed 5.\n4) If 5 is not a power of 2, we multiply the appropriate intermediate results.\n\nLet's do the calculation:\n\n18^1 mod 89 = 18\n18^2 mod 89 = 57\n18^4 mod 89 = 45\n...\n\n(continuing this process until we reach or exceed 5)\n\nAfter performing all these steps, we find that:\n\n18^5 mod 89 = 9\n\nThis means that when we divide 18^5 by 89, the remainder is 9.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Find the greatest common divisor of 624 and 424.",
"role": "human"
},
{
"content": "To find the greatest common divisor of 624 and 424, we can use the Euclidean algorithm. Let's work through it step by step:\n\n1) First, let's set up our initial values:\n a = 624\n b = 424\n\n2) Now, we'll repeatedly divide a by b and update our values until we get a remainder of 0:\n\n 624 = 424 * 1 + 200\n\n We continue this process until the remainder is 0.\n\n3) The last non-zero remainder is our GCD.\n\nAfter performing these steps, we find that the greatest common divisor of 624 and 424 is 8.\n\nThis means that 8 is the largest positive integer that divides both 624 and 424 without leaving a remainder.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Calculate 12^9 modulo 48.",
"role": "human"
},
{
"content": "Let's calculate 12^9 modulo 48. \n\nTo do this efficiently, we can use the concept of modular exponentiation. This allows us to compute large powers without actually calculating the entire number and then taking the modulus.\n\nHere's how it works:\n\n1) We start with the base case: 12^1 mod 48 = 12\n2) Then we square this result and take modulus 48 again.\n3) We repeat step 2, each time doubling the exponent, until we reach or exceed 9.\n4) If 9 is not a power of 2, we multiply the appropriate intermediate results.\n\nLet's do the calculation:\n\n12^1 mod 48 = 12\n12^2 mod 48 = 0\n12^4 mod 48 = 0\n...\n\n(continuing this process until we reach or exceed 9)\n\nAfter performing all these steps, we find that:\n\n12^9 mod 48 = 0\n\nThis means that when we divide 12^9 by 48, the remainder is 0.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Calculate 15! modulo 75.",
"role": "human"
},
{
"content": "Let's calculate 15! modulo 75. \n\nFirst, let's break this down:\n\n1) We need to calculate 15! (factorial of 15)\n2) Then we need to find the remainder when this factorial is divided by 75\n\nStep 1: Calculate 15!\n15! = 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12 * 13 * 14 * 15 = 1307674368000\n\nStep 2: Find 1307674368000 modulo 75\nWe can use the modulo operation to find the remainder when 1307674368000 is divided by 75.\n\n1307674368000 ÷ 75 = quotient of 17435658240 with a remainder of 0\n\nTherefore, 15! modulo 75 is equal to 0.\n\nThis means that when we divide 15! by 75, the remainder is 0.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Calculate 17! modulo 27.",
"role": "human"
},
{
"content": "Let's calculate 17! modulo 27. \n\nFirst, let's break this down:\n\n1) We need to calculate 17! (factorial of 17)\n2) Then we need to find the remainder when this factorial is divided by 27\n\nStep 1: Calculate 17!\n17! = 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12 * 13 * 14 * 15 * 16 * 17 = 355687428096000\n\nStep 2: Find 355687428096000 modulo 27\nWe can use the modulo operation to find the remainder when 355687428096000 is divided by 27.\n\n355687428096000 ÷ 27 = quotient of 13173608448000 with a remainder of 0\n\nTherefore, 17! modulo 27 is equal to 0.\n\nThis means that when we divide 17! by 27, the remainder is 0.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Calculate 17^11 modulo 20.",
"role": "human"
},
{
"content": "Let's calculate 17^11 modulo 20. \n\nTo do this efficiently, we can use the concept of modular exponentiation. This allows us to compute large powers without actually calculating the entire number and then taking the modulus.\n\nHere's how it works:\n\n1) We start with the base case: 17^1 mod 20 = 17\n2) Then we square this result and take modulus 20 again.\n3) We repeat step 2, each time doubling the exponent, until we reach or exceed 11.\n4) If 11 is not a power of 2, we multiply the appropriate intermediate results.\n\nLet's do the calculation:\n\n17^1 mod 20 = 17\n17^2 mod 20 = 9\n17^4 mod 20 = 1\n...\n\n(continuing this process until we reach or exceed 11)\n\nAfter performing all these steps, we find that:\n\n17^11 mod 20 = 13\n\nThis means that when we divide 17^11 by 20, the remainder is 13.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Calculate 3^2 modulo 43.",
"role": "human"
},
{
"content": "Let's calculate 3^2 modulo 43. \n\nTo do this efficiently, we can use the concept of modular exponentiation. This allows us to compute large powers without actually calculating the entire number and then taking the modulus.\n\nHere's how it works:\n\n1) We start with the base case: 3^1 mod 43 = 3\n2) Then we square this result and take modulus 43 again.\n3) We repeat step 2, each time doubling the exponent, until we reach or exceed 2.\n4) If 2 is not a power of 2, we multiply the appropriate intermediate results.\n\nLet's do the calculation:\n\n3^1 mod 43 = 3\n3^2 mod 43 = 9\n3^4 mod 43 = 38\n...\n\n(continuing this process until we reach or exceed 2)\n\nAfter performing all these steps, we find that:\n\n3^2 mod 43 = 9\n\nThis means that when we divide 3^2 by 43, the remainder is 9.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Calculate 4^12 modulo 50.",
"role": "human"
},
{
"content": "Let's calculate 4^12 modulo 50. \n\nTo do this efficiently, we can use the concept of modular exponentiation. This allows us to compute large powers without actually calculating the entire number and then taking the modulus.\n\nHere's how it works:\n\n1) We start with the base case: 4^1 mod 50 = 4\n2) Then we square this result and take modulus 50 again.\n3) We repeat step 2, each time doubling the exponent, until we reach or exceed 12.\n4) If 12 is not a power of 2, we multiply the appropriate intermediate results.\n\nLet's do the calculation:\n\n4^1 mod 50 = 4\n4^2 mod 50 = 16\n4^4 mod 50 = 6\n...\n\n(continuing this process until we reach or exceed 12)\n\nAfter performing all these steps, we find that:\n\n4^12 mod 50 = 16\n\nThis means that when we divide 4^12 by 50, the remainder is 16.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Calculate 15! modulo 48.",
"role": "human"
},
{
"content": "Let's calculate 15! modulo 48. \n\nFirst, let's break this down:\n\n1) We need to calculate 15! (factorial of 15)\n2) Then we need to find the remainder when this factorial is divided by 48\n\nStep 1: Calculate 15!\n15! = 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12 * 13 * 14 * 15 = 1307674368000\n\nStep 2: Find 1307674368000 modulo 48\nWe can use the modulo operation to find the remainder when 1307674368000 is divided by 48.\n\n1307674368000 ÷ 48 = quotient of 27243216000 with a remainder of 0\n\nTherefore, 15! modulo 48 is equal to 0.\n\nThis means that when we divide 15! by 48, the remainder is 0.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Calculate 13! modulo 92.",
"role": "human"
},
{
"content": "Let's calculate 13! modulo 92. \n\nFirst, let's break this down:\n\n1) We need to calculate 13! (factorial of 13)\n2) Then we need to find the remainder when this factorial is divided by 92\n\nStep 1: Calculate 13!\n13! = 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12 * 13 = 6227020800\n\nStep 2: Find 6227020800 modulo 92\nWe can use the modulo operation to find the remainder when 6227020800 is divided by 92.\n\n6227020800 ÷ 92 = quotient of 67685008 with a remainder of 64\n\nTherefore, 13! modulo 92 is equal to 64.\n\nThis means that when we divide 13! by 92, the remainder is 64.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Find the greatest common divisor of 758 and 464.",
"role": "human"
},
{
"content": "To find the greatest common divisor of 758 and 464, we can use the Euclidean algorithm. Let's work through it step by step:\n\n1) First, let's set up our initial values:\n a = 758\n b = 464\n\n2) Now, we'll repeatedly divide a by b and update our values until we get a remainder of 0:\n\n 758 = 464 * 1 + 294\n\n We continue this process until the remainder is 0.\n\n3) The last non-zero remainder is our GCD.\n\nAfter performing these steps, we find that the greatest common divisor of 758 and 464 is 2.\n\nThis means that 2 is the largest positive integer that divides both 758 and 464 without leaving a remainder.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Find the greatest common divisor of 668 and 481.",
"role": "human"
},
{
"content": "To find the greatest common divisor of 668 and 481, we can use the Euclidean algorithm. Let's work through it step by step:\n\n1) First, let's set up our initial values:\n a = 668\n b = 481\n\n2) Now, we'll repeatedly divide a by b and update our values until we get a remainder of 0:\n\n 668 = 481 * 1 + 187\n\n We continue this process until the remainder is 0.\n\n3) The last non-zero remainder is our GCD.\n\nAfter performing these steps, we find that the greatest common divisor of 668 and 481 is 1.\n\nThis means that 1 is the largest positive integer that divides both 668 and 481 without leaving a remainder.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Find the greatest common divisor of 768 and 266.",
"role": "human"
},
{
"content": "To find the greatest common divisor of 768 and 266, we can use the Euclidean algorithm. Let's work through it step by step:\n\n1) First, let's set up our initial values:\n a = 768\n b = 266\n\n2) Now, we'll repeatedly divide a by b and update our values until we get a remainder of 0:\n\n 768 = 266 * 2 + 236\n\n We continue this process until the remainder is 0.\n\n3) The last non-zero remainder is our GCD.\n\nAfter performing these steps, we find that the greatest common divisor of 768 and 266 is 2.\n\nThis means that 2 is the largest positive integer that divides both 768 and 266 without leaving a remainder.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Find the greatest common divisor of 390 and 566.",
"role": "human"
},
{
"content": "To find the greatest common divisor of 390 and 566, we can use the Euclidean algorithm. Let's work through it step by step:\n\n1) First, let's set up our initial values:\n a = 566\n b = 390\n\n2) Now, we'll repeatedly divide a by b and update our values until we get a remainder of 0:\n\n 566 = 390 * 1 + 176\n\n We continue this process until the remainder is 0.\n\n3) The last non-zero remainder is our GCD.\n\nAfter performing these steps, we find that the greatest common divisor of 390 and 566 is 2.\n\nThis means that 2 is the largest positive integer that divides both 390 and 566 without leaving a remainder.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Find the greatest common divisor of 200 and 716.",
"role": "human"
},
{
"content": "To find the greatest common divisor of 200 and 716, we can use the Euclidean algorithm. Let's work through it step by step:\n\n1) First, let's set up our initial values:\n a = 716\n b = 200\n\n2) Now, we'll repeatedly divide a by b and update our values until we get a remainder of 0:\n\n 716 = 200 * 3 + 116\n\n We continue this process until the remainder is 0.\n\n3) The last non-zero remainder is our GCD.\n\nAfter performing these steps, we find that the greatest common divisor of 200 and 716 is 4.\n\nThis means that 4 is the largest positive integer that divides both 200 and 716 without leaving a remainder.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Calculate 8! modulo 11.",
"role": "human"
},
{
"content": "Let's calculate 8! modulo 11. \n\nFirst, let's break this down:\n\n1) We need to calculate 8! (factorial of 8)\n2) Then we need to find the remainder when this factorial is divided by 11\n\nStep 1: Calculate 8!\n8! = 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 = 40320\n\nStep 2: Find 40320 modulo 11\nWe can use the modulo operation to find the remainder when 40320 is divided by 11.\n\n40320 ÷ 11 = quotient of 3665 with a remainder of 5\n\nTherefore, 8! modulo 11 is equal to 5.\n\nThis means that when we divide 8! by 11, the remainder is 5.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Calculate 20^4 modulo 12.",
"role": "human"
},
{
"content": "Let's calculate 20^4 modulo 12. \n\nTo do this efficiently, we can use the concept of modular exponentiation. This allows us to compute large powers without actually calculating the entire number and then taking the modulus.\n\nHere's how it works:\n\n1) We start with the base case: 20^1 mod 12 = 8\n2) Then we square this result and take modulus 12 again.\n3) We repeat step 2, each time doubling the exponent, until we reach or exceed 4.\n4) If 4 is not a power of 2, we multiply the appropriate intermediate results.\n\nLet's do the calculation:\n\n20^1 mod 12 = 8\n20^2 mod 12 = 4\n20^4 mod 12 = 4\n...\n\n(continuing this process until we reach or exceed 4)\n\nAfter performing all these steps, we find that:\n\n20^4 mod 12 = 4\n\nThis means that when we divide 20^4 by 12, the remainder is 4.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Find the greatest common divisor of 262 and 2.",
"role": "human"
},
{
"content": "To find the greatest common divisor of 262 and 2, we can use the Euclidean algorithm. Let's work through it step by step:\n\n1) First, let's set up our initial values:\n a = 262\n b = 2\n\n2) Now, we'll repeatedly divide a by b and update our values until we get a remainder of 0:\n\n 262 = 2 * 131 + 0\n\n We continue this process until the remainder is 0.\n\n3) The last non-zero remainder is our GCD.\n\nAfter performing these steps, we find that the greatest common divisor of 262 and 2 is 2.\n\nThis means that 2 is the largest positive integer that divides both 262 and 2 without leaving a remainder.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Find the greatest common divisor of 708 and 397.",
"role": "human"
},
{
"content": "To find the greatest common divisor of 708 and 397, we can use the Euclidean algorithm. Let's work through it step by step:\n\n1) First, let's set up our initial values:\n a = 708\n b = 397\n\n2) Now, we'll repeatedly divide a by b and update our values until we get a remainder of 0:\n\n 708 = 397 * 1 + 311\n\n We continue this process until the remainder is 0.\n\n3) The last non-zero remainder is our GCD.\n\nAfter performing these steps, we find that the greatest common divisor of 708 and 397 is 1.\n\nThis means that 1 is the largest positive integer that divides both 708 and 397 without leaving a remainder.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Calculate 10^18 modulo 44.",
"role": "human"
},
{
"content": "Let's calculate 10^18 modulo 44. \n\nTo do this efficiently, we can use the concept of modular exponentiation. This allows us to compute large powers without actually calculating the entire number and then taking the modulus.\n\nHere's how it works:\n\n1) We start with the base case: 10^1 mod 44 = 10\n2) Then we square this result and take modulus 44 again.\n3) We repeat step 2, each time doubling the exponent, until we reach or exceed 18.\n4) If 18 is not a power of 2, we multiply the appropriate intermediate results.\n\nLet's do the calculation:\n\n10^1 mod 44 = 10\n10^2 mod 44 = 12\n10^4 mod 44 = 12\n...\n\n(continuing this process until we reach or exceed 18)\n\nAfter performing all these steps, we find that:\n\n10^18 mod 44 = 12\n\nThis means that when we divide 10^18 by 44, the remainder is 12.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Calculate 16^17 modulo 17.",
"role": "human"
},
{
"content": "Let's calculate 16^17 modulo 17. \n\nTo do this efficiently, we can use the concept of modular exponentiation. This allows us to compute large powers without actually calculating the entire number and then taking the modulus.\n\nHere's how it works:\n\n1) We start with the base case: 16^1 mod 17 = 16\n2) Then we square this result and take modulus 17 again.\n3) We repeat step 2, each time doubling the exponent, until we reach or exceed 17.\n4) If 17 is not a power of 2, we multiply the appropriate intermediate results.\n\nLet's do the calculation:\n\n16^1 mod 17 = 16\n16^2 mod 17 = 1\n16^4 mod 17 = 1\n...\n\n(continuing this process until we reach or exceed 17)\n\nAfter performing all these steps, we find that:\n\n16^17 mod 17 = 16\n\nThis means that when we divide 16^17 by 17, the remainder is 16.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Find the greatest common divisor of 848 and 561.",
"role": "human"
},
{
"content": "To find the greatest common divisor of 848 and 561, we can use the Euclidean algorithm. Let's work through it step by step:\n\n1) First, let's set up our initial values:\n a = 848\n b = 561\n\n2) Now, we'll repeatedly divide a by b and update our values until we get a remainder of 0:\n\n 848 = 561 * 1 + 287\n\n We continue this process until the remainder is 0.\n\n3) The last non-zero remainder is our GCD.\n\nAfter performing these steps, we find that the greatest common divisor of 848 and 561 is 1.\n\nThis means that 1 is the largest positive integer that divides both 848 and 561 without leaving a remainder.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Find the greatest common divisor of 249 and 794.",
"role": "human"
},
{
"content": "To find the greatest common divisor of 249 and 794, we can use the Euclidean algorithm. Let's work through it step by step:\n\n1) First, let's set up our initial values:\n a = 794\n b = 249\n\n2) Now, we'll repeatedly divide a by b and update our values until we get a remainder of 0:\n\n 794 = 249 * 3 + 47\n\n We continue this process until the remainder is 0.\n\n3) The last non-zero remainder is our GCD.\n\nAfter performing these steps, we find that the greatest common divisor of 249 and 794 is 1.\n\nThis means that 1 is the largest positive integer that divides both 249 and 794 without leaving a remainder.",
"role": "assistant"
}
] |
[
{
"content": "You are a helpful assistant which is good at math, coding, chess and logic.",
"role": "system"
},
{
"content": "Calculate 1! modulo 74.",
"role": "human"
},
{
"content": "Let's calculate 1! modulo 74. \n\nFirst, let's break this down:\n\n1) We need to calculate 1! (factorial of 1)\n2) Then we need to find the remainder when this factorial is divided by 74\n\nStep 1: Calculate 1!\n1! = 1 = 1\n\nStep 2: Find 1 modulo 74\nWe can use the modulo operation to find the remainder when 1 is divided by 74.\n\n1 ÷ 74 = quotient of 0 with a remainder of 1\n\nTherefore, 1! modulo 74 is equal to 1.\n\nThis means that when we divide 1! by 74, the remainder is 1.",
"role": "assistant"
}
] |
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