# British Maths Olympiad (BMO) 2003 Round 1 Question 5 does a solution with fermat's theorem or crt exist?

The question states:

Let $p,q$ and $r$ be prime numbers.

It is given

$p$ divides $qr − 1$,

$q$ divides $rp − 1$,

$r$ divides $pq − 1$.

Determine all possible values of $pqr$

My specific question is does a solution with fermat's theorem or crt exist? My attempt along those lines follows:

We know that $p,q,r$ cannot all be odd, in-fact exactly one of them must be event. $w.l.o.g$ we let $p = 2$ then we have $q,r$ are odd. This means

$qr - 1$ mod $p = 0$ $\Rightarrow$ $qr$ mod $2 = 1$

$2r - 1$ mod $q = 0$ $\Rightarrow$ $2r$ mod $q = 1$

$2q - 1$ mod $r = 0$ $\Rightarrow$ $2q$ mod $r = 1$

Thus we have $r$ is the inverse modulo $q$ of 2 and similarly for q modulo $r$. Not sure if any of this is useful...

P.S I know this question has been asked on this website but here I am specifically asking if a type of solution exists