# struggling with proof: gcd(k∗a, k∗b)=k∗gcd(a, b)

Any time you are working with gcd, you will probably want to start with terms of the form $kax+kby$. You might even want to consider the set $A = \{kax+kby\in \mathbb{N} \mid x,y \in \mathbb{Z}\}$. The minimum of that set is the gcd. Next show that the right hand side produces a divisor of both ka and ka. Then show it produces a greatest divisor.