Proof : If d= gcd(a,b) , then d is the smallest | Abstract Algebra

This is a little unconventional proof that if d is the gcd of a and b then d is the smallest such natural number.
Also, this is the proof of the following statement from Gallian's Abstract Algebra.
"Let d = gcd(a, b). If a = da` and b = db`, show that gcd(a`, b`) = 1.

If there is any confusion, ask me in the comments.