Please help me prove this question

Question

The Question: We can define a set of integers $X_a$$_,$$_b$ = {∀u, v ∈ $\Bbb Z$, au + bv}. For example, if a = 6 and b = 8 then X$_6$$_,$$_8$ includes numbers like 20 = 2$*$6 + 1$*$8 and 4 = −2$*$6 + 2$*$8. Let c be the smallest positive integer in X$_a$$_,$$_b$. Prove that every number in X$_a$$_,$$_b$ is a multiple of c.

Answer 1

Divide the arbitrary element of $X_{a,b}$ by $c$

The reminder if not zero is a positive number less than $c$ which is also an element of $X_{a,b}$

I let you show that the remainder is indeed an element of $X_{a,b}.$