(==>) Suppose a=b. Then the integer multiples of a are the integer multiples of b, so aZ=bZ

(<==) Suppose aZ=bZ. Since a is a multiple of itself, a is an element of aZ. Since aZ is a subset of bZ, a is an element of bZ. Then a is a multiple of b: a=bp for some integer p. Similarly, since bZ is a subset of aZ, b is a multiple of a: b=aq for some integer q. Then b=aq=(bp)q=b(pq). Since b is not zero, pq=1. Then p=q=1 or p=q=-1, since p and q are integers. It is assumed that a and b are natural numbers, so they have the same sign. This implies p=q=1, so a=b.