Question

If there are integers a, b, s, and t such that, the sum at+bs=1, show that GCD(a,b)=1

Answer 1

We have $at+bs=1$.

Assume that $GCD(a,b)=n$. Then by defenition $n|a$ and i there exists $m|a$ and $m|b$.

Since $n|a$ we have $n|at$ and $n|bs$. Hence $n|(at+bs)$. This implies $n|1$ i.e. $n=1$.