TITLE: Prove that : $(a-b)(a-c)(a-d)(b-c)(b-d)(c-d)$ divisible by $12$
QUESTION [5 upvotes]: Prove that : $(a-b)(a-c)(a-d)(b-c)(b-d)(c-d)$ divisible by $12$, with $a,b,c,d\in\mathbb{Z}$.

REPLY [8 votes]: As drhab pointed out, $a,b,c,d$ must be integers.
Since each of $a,b,c,d$ is either even or odd, there are at least two pairs which have the same parity. Hence, the given value is a multiple of $4$.
Since each of $a,b,c,d$ is either $0,1$, or $2$ in mod $3$, there is at least one pair which is congruent in mod $3$. Hence, the given value is a multiple of $3$.