Looking at the solution of an exercise on Sylow theorems, I see that, if $q$ and $p$ are odd primes such that $q|(p-1),$ then $q$ and $p(p+1)$ are relatively prime.
I understand that $q$ and $p$ are relatively prime, but is there a reason (theorem) why $q$ and $(p+1)$ are relatively prime?
Thank you for your help.
Best Answer
If $d|q$ and $d|(p+1)$ and $q|(p-1),$ then $d|(p-1)$ and $d|(p+1),$ so $ d|2,$ so $d=1$ or $2,$
but $q$ is odd, so $d=1.$ Therefore $q$ and $p+1$ are relatively prime.