I'm trying to prove that the following system of congruence equations has a solution:
$X \equiv 2 $ (mod $5^N$)
$X \equiv 1 $ (mod $7^N$)
$X \equiv 4 $ (mod $6^N-4$)
being $N$ an integer number, $N\geq 2$
I guess that this may be answered using the Chinese Remainder Theorem, for which I need to show that the numbers $5^N$, $7^N$, $6^N-4$ are pairwise coprimes integers. I can see that this is certainly the case between $5^N$ an $7^N$ because $5$ and $7$ are prime integers, but I am not being able to see it between $5^N$ (or $7^N$) and $6^N-4$.
Maybe someone can help me to see why they are coprimes or perhaps, if I am not on the right track, suggest me another way of proving the existence of the solution of the system.