$ \underline {Positive \ \ solution \ \ of \ \ Diophantine \ \ linear \ \ equation : } \ $
Consider the Diophantine equation
$$ px+qy=n \ $$
where $ \ p , \ q \ $ are positive integers and $ \ n \ $ is non-negative integers.
$ \ (a) \ $ Prove that if $ \ gcd(p,q) $ is a factor of $ \ n \ $ , then
(i) If $ \ n>pq-p-q \ $ , there is at least one solution in positive integers $ \ x \ \ and \ \ y \ $ .
(ii) If $ \ n=pq-p-q \ $ , there is no solution in positive integers .
Answer:
$ \ (i) \ $
Let $ \ d=gcd(p,q) \ $ and given that $ \ d \ | \ n \ $.
Now $ \ d \ | n \Rightarrow n=kd \ $ , for some $ \ k \in \mathbb{Z} \ $
Now since $ \ n \ $ is non-negative integer , $ \ k \ $ is also non-negative.
Since $ \ p , q \ $ are positive integers , $ \ gcd(p,q)=d \geq 1 \ $
But now I can not prove the the conclusion of (i)
please get through the solution at least first one.