Question

If $n$ is a prime number greater than $7$, shew that $n^6-1$ is divisible by $504$.

Answer 1

By Fermat's theorem of numbers;-

$n^6-1=M\left(7\right)$

Also $n^6-1$ is divisible by $n^2-1$ and hence by $8$.

Since 'n' must be of the form $3q+1$ or $3q-1$, it is easy to see that $n^4-1$is divisible by $9$, hence the given expression is divisible by $7,8,9$ or $504$