Determine all triplets of integers (a, b, c) satisfying 1 < a < b < c such that abc - 1 is a multiple of (a - 1)(b -1)(c - 1)
Analysing the variables concerning whether they are even or odd, the following possibilities exist (there is no distinction between a, b and c):
a b c abc-1 (a-1)(b-1)(c-1)
e e e o o
e e o o e
e o o o e
o o o e e
x(a-1)(b-1)(c-1) must have the same oddity as abc-1, but an even number can not become an odd one by multiplication, so cases 2 and 3 are discarded. The only remaining are: e e e and o o o. So all a, b and c must be either even or odd.
Some more analysis
