If $pq=1$, then $p=q=1$ for $p,q \in \mathbb {Z}, p,q >0$

Question

If $pq=1$, then $p=q=1$ for $p,q \in \mathbb {Z}$, $p,q >0$

I tried to do this by contradiction and I get

$(pq=1) \land (p\neq 1 \lor q \neq 1)$

then I have no ideas how to continue with a formal proof. What I know is if i choose q not equal to 1 and it is greater then 0, I will get

$p = \frac{1}{q}$

and p will be in the interval (0-1) and I get a contradiction.

Do I have to do (1) if p is not 1 and (2) both p and q is not 1?

I don't know what tag is appropriate here, but real-analysis is surely not. — Mariano Suárez-Álvarez, May 12 '13 at 03:05

score 2 · Accepted Answer · answered May 12 '13 at 03:04

2

Suppose $p$ and $q$ are two positive integers such that $pq=1$.

If $p>1$, then $q=1/p<1$ and this is impossible because $q$ is a positive integer. As $p$ cannot be less than $1$, again because it is a positive integer, it can only be equal to $1$. Of course, $q$ is also $1$ in that case.

answered May 12 '13 at 03:04

Mariano Suárez-Álvarez

135,076

So I have to check for all cases? – Timothy Leung May 12 '13 at 03:09
What «all cases»? Are you looking for a proof shorter than two short lines?! :-) – Mariano Suárez-Álvarez May 12 '13 at 03:13
(1) p > 1 (2) q>1 (3) both p, q > 1 – Timothy Leung May 12 '13 at 03:14
No: my proof is complete. As soon as you know that $p$ is one you of course also know that $q$ is also $1$. – Mariano Suárez-Álvarez May 12 '13 at 03:14

If $pq=1$, then $p=q=1$ for $p,q \in \mathbb {Z}, p,q >0$

1 Answers1