Theorem:

For any natural number m, gcd(ma,mb) = m gcd(a,b)

Proof:

Let N={natural numbers}

By another theorem, we know that

gcd(ma,mb)=min({m(ax)+m(by): x,y E Z} ∩ N)

=m * min({ax+by: x,y E Z} ∩ N)

=m * gcd(a,b)

===========================

I don't understand the step

min({m(ax)+m(by): x,y E Z} ∩ N) = m * min({ax+by: x,y E Z} ∩ N).

Why is this true? Can someone please explain this?

Thanks for any help!