I encountered some other problem and I found a beautiful proof here Write $1/1 + 1/2 + ...1/ (p-1)=a/b$ with $(a,b)=1$. Show that $p^2 \mid a$ if $p\geq 5$. (see Thomas Andrew's post)
But I thought he may miss the proof that $(a_1,(p-1)!)=1,$ which is not trivial for me. (As for the definitions of $a_1$ and $p$ please see the link above. The definitions are simple and clear there.)
My problem is just that how to prove $(a_1,(p-1)!)=1$.
I tried to use the property that $(n,m)=(n.n+km)$ for any integers $n,m,k.$ But it turns out it makes the expression messy and dirty. And I can't go further.
Any help will be thanked.