Divisibility of polynomial; basic number theory

Question

Show that $42|(n^7 - n)$ for all integers $n$

The only hint I've been given uses Fermat's Little Theorem but I don't know how to use that in this case since $42$ is not a prime.

you can write $42=2\cdot 3\cdot 7$ – Yanko Jan 27 '18 at 14:36 — Yanko, Jan 27 '18 at 14:36
At least you could have viewed my edit :( – Yash Jain Jan 27 '18 at 14:37 — Yash Jain, Jan 27 '18 at 14:37

José Carlos Santos · Accepted Answer · 2018-01-27T16:59:36.700

2

Well, $42=2\times3\times7$ and:

$n$ and $n^7$ have the same parity, which means that $2\mid n^7-n$;
by Fermat's little theorem, $3\mid n^3-n$, and$$n^7-n=n^7-n^5+n^5-n^3+n^3-n=(n^4+n^2+1)(n^3-n),$$which implies that $3\mid n^7-n$;
by Fermat's little theorem, $7\mid n^7-n$.

Since $2$, $3$, and $7$ are co-prime, this implies that $42\mid n^7-n$.

edited Jan 27 '18 at 16:59

answered Jan 27 '18 at 14:43

1 Answers1