Hello Im having some trouble with this proof Can someone ple

Hello, I\'m having some trouble with this proof. Can someone please help me? Thanks

Let c, d, f and g be integers. If gcd(f, g) = d and c > 0, then gcd(fc, gc) = dc.

Solution

Let a and b be integers, not both 0. Then gcd(a,b) is the smallest positive integer which can be expressed in the form au+bv, where u and v are integers.

Let d=gcd(a,b), and let w=gcd(ax,bx). Suppose that x is positive.

By Bézout\'s lemma, w is the smallest positive integer which can be expressed in the form w=(ax)u+(bx)v. From this last equation, we can see that x divides w, so w=xk for some k. It follows that k=au+bv. We will show k=d.

By Bézout\'s lemma, d is the smallest positive integer which can be expressed as an integer linear combination of a and b. Since k=au+bv, we conclude that dk.

There are integers s and t such that as+bt=d. It follows that (ax)s+(bx)t=xd. Since w=xk is the smallest integer that is an integer linear combination of ax and bx, we conclude thatxkxd, and therefore kd.

We have shown that dk and kd. Thus k=d, and therefore gcd(ax,bx)=xgcd(a,b).