Past exam of the mathematics course of the University of Cambridge / 2015 / ia / Paper 4 / 1E / b / Solution

Write the composite number as $n=ab$, with $2\le a\le b<n$. If $a<b$, both $a$ and $b$ occur as distinct factors of the factorial $(n-1)!$, so their product $n$ divides that factorial.

If $a=b$, then $n=a^2$ and $n>4$ implies $a\ge 3$. Both $a$ and $2a$ are distinct factors in $(n-1)!$, because $2a\le a^2-1$. Their product is $2n$, so again $n$ divides the factorial. HenceThe exception $n=4$ matters: its factorial $3!=6$ is not divisible by $4$. This argument handles perfect squares without incorrectly counting the same factor twice.