Question 47
Let $n \gt 6$ be a perfect number. Let $p_{1}^{a_{1}} \cdot p_{2}^{a_{2}} \cdot \ldots \cdot p_{k}^{a_{k}}$ be the prime factorisation of $n$ where we assume that $p_{1} \lt p_{2} \lt \ldots \lt p_{k}$ and $a_{i} \gt 0$ for all $i = 1, \ldots, k$. Prove that $a_{1}$ is even.