Find gcd of $\displaystyle 7^{4n+3}-4^{10n+3}+11^{10n+2}$, $\displaystyle n \in N_0$.
Any idea?
The second term is always divisible by 8, so the problem reduces to investigate whether $\displaystyle 7^{4n+3}+11^{10n+2}$ is divisible by 8 or not.
Since $\displaystyle 7^{2}$ is 1 modulo 8 we have that $\displaystyle 7^{4n}$ is also 1 modulo 8. Therefore $\displaystyle 7^{4n+3} \equiv 7 \equiv -1 \mod 8$.
On the other hand, the fact that $\displaystyle 11^{2} \equiv 1 \mod 8$ implies at once that $\displaystyle 11^{10n+2} \equiv 1 \mod 8$. Hence $\displaystyle 7^{4n+3}+11^{10n+2} \equiv (-1)+1 = 0 \mod 8$ as was to be shown.