Coprime Triples

Number Theory Level 5

Find the number of pairwise coprime triples of positive integers $(a,b,c)$ with $a < b < c$ such that

$a| bc-31, b|ca-31 \mbox{ and }c|ab-31.$

Details and assumptions

The notation $n \mid m$ means that $n$ is a divisor of $m$ .

Clarification: $(ab-31), (ac-31),$ and $(bc-31)$ may be zero or negative.