Prove that every multiple of some integer b is a multiple of another integer a if, and only if b divides a. This was one of homework problems in Introduction to Abstract Algebra I was in for two lectures.
Let denote the set of all integer multiples of a, denote the set of all multiples of b.
1. Prove that implies
Suppose , then for some integer k by the definition of divisibility. Let so that for some integer m by the definition of . Then by substitution. Hence by associative law. Thus as desired since .
2. Prove that implies
Suppose for every e in , e is also in . By the definition of , for some integer k for every e in . Similarly, for some integer m since is a subset of . Then . In particular, if we choose , then for some integer n. Hence as desired.
Since 1 and 2 are both proven, we have shown that if and only if . Q.E.D.