Let $a$ and $b$ be two distinct positive integers with the same parity. Prove that the fraction $\frac{a!+b!}{2^a}$ is not an integer.
Source: Romania JBMO TST 2023 Day 4
Tags: number theory
Let $a$ and $b$ be two distinct positive integers with the same parity. Prove that the fraction $\frac{a!+b!}{2^a}$ is not an integer.