how to prove Any element c $\in$ L is a multiple of m. where m is the min of L

Question

Let $a, b \in \mathbb N$, assume they are not both $0$. Define $L = \{n\in\mathbb N^+ \mid \exists x, y \in \mathbb{Z}: n = ax + by\}$

how do I prove the following claim:

Any element c $\in$ L is a multiple of m. Where m is the smallest element of L

Answer 1

Suppose the claim is false. Let $L\ni c=dm+e,e<m$. By definition, $\exists x_1,y_1,x_2,y_2\in\mathbb Z$, $$m=ax_1+by_1$$ $$dm+e=c=ax_2+by_2$$ Hence, $$e=a(x_2-dx_1)+b(y_2-dy_1)$$ Since $x_2-dx_1,y_2-dy_1\in\mathbb Z$, $e\in L$, but $e<m$, contradicts to the fact that $m$ is the smallest element in $L$.