2014/025) If p, q and r are prime, and p divides qr − 1, q divides rp − 1, and r divides pq − 1, find all possible values of pqr

all 3 are different because if p = q then p divides pr-1 and

as p divides pr so p divides 1. so p is not prime. Without loss of generality we can assume p < q < r.

Pqr divided (pq-1)(qr-1)(rp-1)

so (pq-1)(qr-1)(rp-1)= mpqr where m is integer

so pqr – (pq^2r+p^2rq + pqr^2) + (pq + qr + rp) – 1 = mpqr

or pqr(1-p-q-r -m) = 1- (pq + qr + rp)

or pqrn = (pq + qr + rp) – 1 where n = p+q+r+m – 1

or dividing by pqr we get

1/ p + 1/q + 1/r = n + 1/(pqr)

as 1/p > 1/(pqr) we have n > 0

as 1/ p + 1/q + 1/r < 3 * (1/2) so n < 3/2

or n = 1

so 1/p + 1/q + 1/r = 1 + 1/(pqr) or > 1 ... (1)

p > 1 and p < 3 as 3 * 1/3 = 1 and so if p = 3 then 1/p + 1/ q + 1/r < 1

so p = 2 and q < 4 as 1/2 + 1/4 + 1/5 < 1

so q = 3

from (1) we get p = 2, q =3, r = 5 or pqr = 30