For any $p\leqslant n$($ p $ is a prime),we have $$v_p(a^{n!}-1)=v_p(a^{p-1}-1)+v_p(\frac{n!}{p-1})>v_p(n!)$$

Also, if $n!=\textstyle \sum_{1\le k\le L}p_k^{e_k}$ for primes $p_1<\cdots<p_L\le n$, then note that $p_i^{e_i-1}\mid n!$ and $p_i-1\mid n!$ (as $p_i-1<n$), so that $\varphi(p_i^{e_i})\mid n!$ and by Euler's theorem, $p_i^{e_i}\mid a^{n!}-1$. Since $p_i$ are pairwise coprime, the conclusion follows.