TITLE: Arithmetics ; $p-1 \mid q$ equivalent to $(p-1)^2 \mid p^q - 1$
QUESTION [5 upvotes]: I have a problem with this exercise :
prof that  $p-1|\space q  \iff (p-1)^2|\space p^q - 1$
I succeed to prof that $(p-1)^2|\space p^q - 1 \implies p-1|\space q$ 
thanks ^^

REPLY [4 votes]: $\rm{\bf Hint}\:\ (p\!-\!1)^2\! \mid p^q\!-1 \!\iff\! p\!-\!1\ \bigg|\ \dfrac{p^q\!-1}{p\!-\!1} = p^{q-1}\! +\cdots\!+p\! +\! 1\equiv 1+\cdots\!+1$ $\rm\equiv q\ (mod\ p\!-\!1)$
As I explained it has a nice conceptual view as a special case of a derivative test for multiple roots.