If $n$ is composite, prove that $(n-1)!+1$ is not a power of $n$.
Hint: We know that if $n$ is composite and $n>4$ then $(n-1)!$ is divisible by $n$.
Since $n=4$ is the first composite number. We have $(4-1)!+1=7$. Clearly 4 does not divide 7. Also we know that $(n-1)! \equiv 0$ (mod $n$) (for $ n>4$ and $ n$ composite). Also $1 \equiv 1$ (mod $n$). Adding both these equations we get :
$(n-1)!+1 \equiv 1$ $\pmod n.$ Hence it is clear that $(n-1)!+1$ is not a power of $n$.
Please correct me if there is any discrepancy in proof writing or the solution. Also it is highly appreciable if someone could provide with any other solution (Using modular-arithmetic or using Wilson's Theorem).
Thanks in advance.