Me and a friend are having a small arguement. My friend says the statement is false but I'm saying its true.
My friend thinks its false because ka + lb does not necessarily equal 1. Rather, it can equal anything as long as k and l holds.
I say its true because the gcd(a,b) = d = ka + lb, a theorem we used earlier.
We understand this is supposed to be easy, but we can't agree on this.
One can show the ideal $I=\{ka+lb\mid k,l\in\mathbf Z\}$ is equal to the ideal generated by $\gcd(a,b)$. Hence if $ka+lb=1$ for some $k,l\in \mathbf Z$, $a$ and $b$ are coprime.
Indeed, if $d=k_0 a+l_0 b$ is the smallest positive element of $I$, you can perform a Euclidean division of $a$ by $d$: $a=qd+r\enspace(0\le r<d)$. Then $$0\le r=a-qd=(1-qk_0)a -ql_0b$$ is a non-negative element of $I$ which is smaller than $d$. By the minimality of $d$, we have $r$, i. e. $d$ divides $a$. For similar reasons, $d$ divides $b$.