Divisibility using Congruence

Question

Is $11^{11296}-1$ divisible by 11 and/or 12? It is easy to see that is divisible by 10 as $x^n-1$ has factor $x-1$. I can't see how to apply Fermat's or Eulers theorem of congruence (if applicable)? What would be the approach to test divisibility by 11 and/or 12?

Answer 1

Divisibility by $11$: it's not, because we know $11^{11296}$ is divisible by $11$, so subtracting $1$ gives a number that is not divisible by $11$.

Divisibility by $12$: $11 \equiv -1$ (mod $12$) so $11^{11296} \equiv (-1)^{11296} $ (mod $12$). This is equivalent to $1$ and therefore subtracting $1$ gives you $0$. This means the number is divisible by $12$ as it is equivalent to $0$ in a mod $12$ system.