Math Gold Medalist

Lor

2023 AIME I 

Problem 4

The sum of all positive integers $m$ such that $\frac{13!}{m}$ is a perfect square can be written as $2^a3^b5^c7^d11^e13^f,$ where $a,b,c,d,e,$ and $f$ are positive integers. Find $a+b+c+d+e+f.$


Legendre’s Formula

Idea of Sum of divisors

Important Identities

Prime factorize 13!

Power of each prime in 13!/m should be even

Find possible cases of m

Use the idea of sum of positive divisors

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   Solution