Exercise 2.1.25 ($n^{12} - a^{12}$ is divisible by $91$ if $n$ and $a$ are prime to $91$.)

Proof. Suppose that $n$ and $a$ are prime to $91$. The decomposition of $91$ in prime factors is $91=13\times 7$.

Therefore $n$ and $a$ are prime to $13$. By Problem 24,

Moreover $n$ and $a$ are prime to $7$, thus $7\mid n^6-1$ and $7\mid a^6-1$, so $7\mid n^6-a^6$. Moreover $n^{12}-a^{12}=(n^6-a^6)(n^6+a^6)$, thus

Since $7\wedge 13=1$,

$n^{12}-a^{12}$ is divisible by $91$ if $n$ and $a$ are prime to $91$. □