Sum of multiple of two prime numbers

Question

Let $a$ and $b$ be two prime numbers. Which numbers are in this set?

$\{ax + by | x,y \in \Bbb {Z}\}$

Answer 1

If $a \ne b$ then from Bezout's identity, any number can be represented by $ax+by$. Thus, $\mathbb{Z}= \{ ax+by | x,y \in \mathbb{Z} \}$.

If $a=b$ then $\{ ax+by | x,y \in \mathbb{Z} \}= \{ ax | x \in \mathbb{Z} \}$.

Answer 2

What is the $\gcd(a, b) $? What do you know of the Euclidean Algorithm to find the gcd between two numbers?

Remember that the Euclidean Algorithm, to find $\gcd(a, b) $, not only returns $d$, the greatest common divisor, but also an $x_0$ and a $y_0$ in $\Bbb {Z} $ such that

$$x_0a + y_0b = d $$