CH-SMO 2022 Problem 7

Let $n > 6$ be a perfect number. Let $p_1^{a_1} \cdot p_2^{a_2} \cdot ... \cdot p_k^{a_k}$ be the prime factorisation of $n$ , where we assume that $p_1 < p_2 <...< p_k$ and $a_i > 0$ for all $i = 1,...,k$ . Prove that $a_1$ is even.

Remark: An integer $n \ge 2$ is called a perfect number if the sum of its positive divisors, excluding $n$ itself, is equal to $n$ . For example, $6$ is perfect, as its positive divisors are $\{1, 2, 3, 6\}$ and $1+2+3=6$ .