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The smallest positive $ n $ occurs when $ m = 0 $, giving $ n = 16 $. Verifying: $ 16 \div 16 = 1 $ and $ 16 - 1 = 15 $, which is divisible by 5.
Question: What is the sum of all the squares of the divisors of $ 12 $?
Solution: The divisors of $ 12 $ are $ 1, 2, 3, 4, 6, 12 $. Compute the sum of their squares:
1^2 + 2^2 + 3^2 + 4^2 + 6^2 + 12^2 = 1 + 4 + 9 + 16 + 36 + 144 = 210
Question: What is the largest integer that must divide the product of any five consecutive integers in a neural network's orbital cycle?
Solution: The product of $ k $ consecutive integers is divisible by $ k! $. For five consecutive integers, this is $ 5! = 120 $. For example, $ 1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 = 120 $, and $ 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 =
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