Find all pairs $(a,b)$ of positive integers, such that $(a+1)(b+1)$ is divisible by $ab$.

Question

Problem: Find all pairs $(a,b)$ of positive integers, such that $ab\mid (a+1)(b+1)$. The solution from the source begins like this: We have $(a+1)(b+1)=ab+a+b+1$, thus $a\mid b+1$ and $b\mid a+1$.(Why?)

Answer 1

We want to have $ab\mid (a+1)(b+1)$. That implies $a\mid (a+1)(b+1)$. Inserting $(a+1)(b+1)=ab+a+b+1$, we get that $a\mid (ab+a+b+1)$. However, we already know that $a\mid ab+a$, so that means that $a\mid b+1$ as well. Similarly for $b$.

Now that we've proven the hint, we get that $a\leq b+1$ and $b\leq a+1$. Say that $a\leq b$. Then we have either $a=b$ or $a+1=b$. In the first case we get $a\mid a+1$, so $\boxed{a=b=1}$ is a solution (as soon as you've checked it). In the second case we get $a\mid a+2$, so either $a=1$ or $a=2$. That means that you need to check $(1,2)$ and $(2,3)$ and see whether they are solutions, and then you're done.