Determine the number of pairs $(a,b) \in [(1,4)]$ which satisfy $|a|,|b|≤40$

Question

Define an equivalence relations on $Z\times(Z-\{0\})$ by letting $(a,b)\sim (c,d)$ if $ad=bc$. Determine the number of pairs $(a,b)\in\mathbb[(1,4)]$ which satisfy $|a|,|b|\leq40$.

Answer 1

Hint: if $(a,b) \in [(1,4)]$ and $a=3$ what choices do you have for $b$? Since $|b| \gt |a|$ for all of these pairs...

Answer 2

Hint: Let $c=1$ and $d=4$. The pair $(a,b)$ is in our equivalence class precisely if $4a=b$ (and $b\ne 0)$. Make a list, and count. Don't forget to include $(1,4)$!

And don't forget about the negative numbers. Note for example that $(-2,-8)$ is in our equivalence class. But there is a useful shortcut. If you have made a careful count of the pairs $(a,b)$ where $a$ and $b$ are positive, double that to get the full count.