(a-b)(a-c)(a-d)(b-c)(b-d)(c-d) -= 0 (mod 3) Can anybody help me with this discrete mathematics question?

Given that a,b,c,d in ZZ , show that
(a-b)(a-c)(a-d)(b-c)(b-d)(c-d) -= 0 (mod 3)

1 Answer
May 2, 2018

See explanation...

Explanation:

What are the values of a, b, c, d modulo 3 ?

They can only take values congruent to 0, 1 or 2.

So by the pigeonhole principle, at least two of a, b, c, d are congruent modulo 3.

Then their difference is a multiple of 3 and hence:

(a-b)(a-c)(a-d)(b-c)(b-d)(c-d) -= 0 (mod 3)