Greatest Common Divisor Semigroups

Question

I'm trying to find a proof that shows if $a$, $b$ are in the natural numbers, then the sum of the additive semigroups $\mathbb N a + \mathbb N b$ is a subset of $\mathbb N d$ where $d = \gcd(a,b)$.

Answer 1

$d$ is a divisor of both $a$ and $b$. So \begin{align} a & = jd \\ b & = kd \end{align} for some $j,k\in\mathbb N$.

So $ma+nb = mjd + nkd = (mj+nk)d \in d\mathbb N$.