MHB Is the Product \(abc(a^3-b^3)(b^3-c^3)(c^3-a^3)\) Divisible by 7?

[kaliprasad]

Show that for integer a,b,c the product $abc(a^3-b^3)(b^3-c^3)(c^3-a^3)$ is divisible by 7

 

[castor28]

Show that for integer a,b,c the product $abc(a^3-b^3)(b^3-c^3)(c^3-a^3)$ is divisible by 7

Spoiler

If $a$, $b$, or $c$ is divisible by $7$, we are done.

Otherwise, each of $a^3$, $b^3$, $c^3$ is congruent to $\pm1\pmod{7}$. Therefore, at least two of these integers are congruent $\pmod7$, and the corresponding factor is divisible by $7$.