Suppose and are not divisible by , then the left hand side of:
is congruent to modulo while the right hand side cannot be congurent to modulo , a contradiction, hence at least one of and is divisible by .
If both and are divisible then the left hand side is divisible by but the right hand side cannot be divisivle by because is a primitive Pythagorean triple and so share no common factor. This is a contradiction, hence both and cannot be divisible by .
Hence exactly one of is divisible by .
(note we have not used the fact that is even)