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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

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It is unlikely that there would be any meaningful solutions using these theorems. Fermat's little theorem would only be useful in situations involving powers, but this question contains none. Furthermore CRT would produce an infinite number of possibilities based on a modulo and the bounding argument would still be required to fix these.

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