A previous professor I had last year sent me this question, which he says he found an answer but it's too long, and asked me to try and solve it; though after solving it, my answer was a little long, so I'm curious to see if there's any shorter solution. The problem is the following:
Given $p,q$ positive coprime numbers, prove that there exists $a$, $b$ positive integers such that $ap+bq=pq-1$.
I began seeing that by Bezout's identity there exist $A,B\in\mathbb{Z}$ such that $Ap+Bq=1$, then multiplying that by $pq-1$ you get the following: $$A(pq-1)p+B(pq-1)q=pq-1$$ But we don't know if $A(pq-1)$ and $B(pq-1)$ are positive, so what I managed to do was to try to find positive solutions starting from those. What I did was set a general solution (of the equation $a'p+b'q=pq-1$) using that initial solution, and try to find a certain parameter $k$ which could make the solutions, let's call them $A_k$ and $B_k$ both positive; I hope this is enough explanation, otherwise I can try to explain more what I did but it would take me quite some time.
