Is it true that $\gcd(p+1,pk)=1$.

Question

Let $p$ be a prime. Consider the statement $\gcd(p+1,pk)=1$. The statement does not seem to be true for all $k$. In particular, take $k=3,p=2,\gcd(2+1,2(3))=3>1$. So is this statement true if $\gcd\nmid k$?

Answer 1

HINT: gcd$(p+1,pk)=$ gcd$(p+1,k)$.

To see this, let $q$ be a prime and $r$ a positive integer such that $q^r$ divides $(p+1)$ and $q^r$ divides $pk$. Then as $p$ does not divide $p+1$ it follows that $q \not =p$ so $q$ does not divide $p$ and so $q^r$ must divide $k$. So if $q^r$ divides $(p+1)$ and $q^r$ divides $pk$ then $q^r$ must divide both $p+1$ and $k$.