RTP: If , then , where a,b are positive integers

Proof

Note

Assume

Then

Thus

So

So

This is impossible, so our assumption is wrong

Thus our statement is true

Questions

1. Is this proof valid?

2. Can I generalize to :If gcd(a,b)=k, then gcd(a, (a+b))=k

3. Is there a simpler non-contradiction, non-FTA proof?