Show that if a b element Z and c element N such that a Ident

Show that if a, b element Z, and c element N such that a Identical b (mod c), then gcd (a, c) = gcd(b, c).

a=b mod c

hence, a-b|c

a-b=kc where k is some integer

Let, gcd(a,c)=p, gcd(b,c)=q

So , a=pm,c=pn

pm-b=kpn

Hence, p|b and p|c

But, q is gcd(b,c) hence, q|p

Similarly we can show:p|q

Hence, p=q

HEnce proved