2023 AIME I Problems/Problem 4
Problem 4
Unofficial problem: The sum of all positive integers such that is a perfect square can be written as , where and are positive integers. Find
Solution
Solution 1
We first rewrite 13! as a prime factorization, which is For the fraction to be a square, it needs each prime to be an even power. must contain . can contain any even power of 2 up to 10, any odd power of 3 up to 5, and any even power of 5 up to 2. The sum of is $(7\cdot11\cdot\13)(2^0+2^2+2^4+2^6+2^8+2^{10})(3^1+3^3+3^5)(5^0+5^2)$ (Error compiling LaTeX. Unknown error_msg), which is .