For what values of a does ax (mod b) take all values (0 to b-1) for all x?

Question

Given fixed positive integers a and b, I know y= ax (mod b) can range from 0 to b-1 for all non negative integers x.

It will always take value 0 (at x=0).

If b is multiple of a, then y will take only values multiples of a(mod b).

My guess is that y will take all values that are multiple of gcd(a,b) mod b

Is this true? How can I prove this?

Thanks