Prove that for any integer n and any integer a, gcd(a, a+n) divides n; hence, gcd(a, a+1)=1

Let g=gcd(a, a+n)a=gc, a+n=gd, where gcd(c,d)=1put a=gc into a+n=gd, we have gc+n=gdthen g(d-c)=n, that is, g divides n.

What is the inequality?