Question

Integers ar coprime if and only if there exist integers x and y such that ax by=1

Answer 1

I know this Bezout's Identity and I saw another question that showed two proofs (one by induction). But I still don't understand them, and I was hoping someone could break them down even further.

My first attempt was:

Proof:

Suppose a and b are relatively prime. The gcd(a,b)=d

therefore d∣a and d∣b

so a=dm and b=dk for some integers m,k

a+b=dm+dk

a+b=dl for some integer l by closure

and then I don't know where to go. Eventually I wanted to get to (a,b)=1 because they are relatively prime and tie that into what I had above.