## Addition of numbers relatively prime

RTP: If $gcd(a,b)=1$, then $gcd(a, (a+b))=1$, where a,b are positive integers

Proof
Note $1=as+bt$
Assume $gcd(a, (a+b))=k$
Then
$am+(a+b)n=a(m+n)+bn=k*1$
$a(m+n)+bn=aks+bkt$
Thus
$m+n=ks,n=kt$
So
$m+kt=ks,m=ks-kt$
So
$a(ks-kt)+bkt=aks+bkt$
$-akt=0$
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?