VMO 2013 Problem 7

Find all ordered 6-tuples satisfy following system of modular equation: $ab+a'b' \equiv 1$ (mod 15) $bc+b'c' \equiv 1$ (mod 15) $ca+c'a' \equiv 1$ (mod 15) Given that $a,b,c,a',b',c' \epsilon (0;1;2;...;14)$