Exactly one of ab, bc and ca is odd => Two are odd and one is even
abc is a multiple of 4 => the even number is a multiple of 4
The arithmetic mean of a and b is an integer => a and b are odd
and so is the arithmetic mean of a, b and c. => a+ b + c is a multiple of 3
c can be 4 or 8.
c = 4; a, b can be 3, 5 or 5, 9
c = 8; a, b can be 3, 7 or 7, 9
Four triplets are possible.