Prove that 3 divides $a \times b \times ( a^2 − b^2 )$.

Question

This is so random and I can't get to the key. All I'm thinking about is to proof that $a \times b \times ( a^2 - b^2 )$ is three numbers that are followed by each other but I can't prove it.

Answer 1

You can write $$ a b (a^2 - b^2) = a b (a+b) (a-b).$$ If $a$ or $b$ is divisible by 3, we're done. If $a$ and $b$ both leave remainder 1 or 2 when divided by 3, $a-b$ is divisible by 3, and if $a$ leaves remainder 1 and $b$ leaves remainder 2 when divided by 3 (or vice versa), $a+b$ is divisible by 3.