So I think I have a solution to this problem but I'm not certain if it works or not.
Problem Statement:
Let a, b, d be non-zero integers. If d|a and d|b, show that d|gcd(a,b).
Attempted Proof
Let $\gcd(a,b) = k$ for some $k$ in integers.
By Bezout's Identity, we can write this as $ax + by = k$ for some $x, y$ in integers.
We know $d|a$ and $d|b$; this implies that $d|ax$ and $d|by$
Since this means d divides all terms on the left side, it divides the entire left side:$ d|ax + by$
This implies $d|k$ which implies $d|\gcd(a,b)$.
Could someone please verify if my solution is valid or not?
