# Thread: the possible values of gcd(a; a + prime).

1. ## the possible values of gcd(a; a + prime).

If p is a prime, find the possible values of gcd(a, a + p).

and find a necessary and sufficient condition for gcd(a, a+p) = 1 where p is a
prime.

2. If $d$ divides both $a$ and $a+p$ then $d|p$, so $d=1$ or $d=p$.

A necessary and sufficient condition that $\mbox{gcd }(a,a+p)=1$ is that $p\nmid a$.