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
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. This means
must contain
. Also,
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
, which simplifies to
.
~chem1kall
Solution 2
The prime factorization of is
.
To get
a perfect square, we must have
, where
,
,
.
Hence, the sum of all feasible is
Therefore, the answer is
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)