Prove that $n^{12} - a^{12}$ is divisible by $91$ if $n$ and $a$ are co prime to $91$
What I tried:
Suppose $n,a$ are integers, relatively prime to $91$.
Since $91=7\cdot 13$, it follows that $n,a$ are relatively prime to $7$ and $13$.
Applying Fermat's little Theorem with the prime $13$, we get \begin{align*} &\begin{cases} n^{12}\equiv 1\;(\text{mod}\;13)\\[4pt] a^{12}\equiv 1\;(\text{mod}\;13)\\ \end{cases}\\[4pt] \implies\;&n^{12}-a^{12}\equiv 0\;(\text{mod}\;13)\\[4pt] \implies\;&13{\,{\mid}\,}\bigl(n^{12}-a^{12}\bigr)\\[4pt] \end{align*} and applying Fermat's little Theorem with the prime $7$, we get \begin{align*} &\begin{cases} n^{6}\equiv 1\;(\text{mod}\;7)\\[4pt] a^{6}\equiv 1\;(\text{mod}\;7)\\ \end{cases}\\[4pt] \implies\;&\begin{cases} n^{12}\equiv 1\;(\text{mod}\;7)\\[4pt] a^{12}\equiv 1\;(\text{mod}\;7)\\ \end{cases}\\[4pt] \implies\;&n^{12}-a^{12}\equiv 0\;(\text{mod}\;7)\\[4pt] \implies\;&7{\,{\mid}\,}\bigl(n^{12}-a^{12}\bigr)\\[4pt] \end{align*} hence, since $7{\,{\mid}\,}\bigl(n^{12}-a^{12}\bigr)$ and $13{\,{\mid}\,}\bigl(n^{12}-a^{12}\bigr)$, it follows that $91{\,{\mid}\,}\bigl(n^{12}-a^{12}\bigr)$.
As regards your proof attempt, the problem doesn't ask you to prove that $n$ is relatively prime to $91$.
It's given that $n,a$ are relatively prime to $91$, and the goal is to prove that $91{\,{\mid}\,}\bigl(n^{12}-a^{12}\bigr)$.
Note that in the first line of your proof attempt, you assume what you are required to prove, and of course, that's not a legal assumption.
Below put $\rm\: n=7\cdot 13,\ f=12$
Theorem $\ $ If $\rm\ n\in \mathbb N\ $ has prime factorization $\rm\:n = p_1^{e_{\:1}}\cdots\:p_k^{e_k}\ $ and $\rm\ \phi(p_i^{e_{\:i}})\ |\ f\,$ for all $\,\rm i,\,$ then $\rm\ n\ |\ a^f-b^f\ $ for all $\rm\: a,b\in \mathbb Z\,$ coprime to $\rm\,n$.
Proof $\ $ By hypothesis $\rm\:a\:$ and $\rm\:b\:$ are coprime to $\rm\: p_i\:$ thus by Euler's phi theorem: $\rm\ mod\ q = p_i^{e_{\:i}}\!:\: \ a^{\phi(q)}\equiv 1\equiv b^{\phi(q)}\: \Rightarrow\ a^f\equiv 1\equiv b^f\, $ so $\,\rm a^f-b^f\equiv 0,\,$ by $\rm\: \phi(q)\ |\ f\:.\ $ Therefore, since all $\rm\ p_i^{e_{\:i}}\ |\ a^f - b^f\\ $ so too does their lcm = product = $\rm\: n\:.\:$
Remark $ $ See here for a generalization for all $\rm\,a,b\,$ (i.e. not necessarily coprime to $\,\rm\,n)$
Two things should leap to mind. And a third thing comes with practice.
1) $91 = 13*7$ so to prove $91|M$ it will work to prove a) $7|M$ and b) $13|M$. If you can prove a) and b) you are done.
2) $n^{12} - a^{12} = (n^6 - a^6)(n^6 + a^6) = (n^3 - a^3)(n^3 + a^3)(n^6 + a^6)=....$. The expression $n^{12}-a^{12}$ is "very" factorable. If we can prove that of the many factors that $7$ and/or $13$ must divide at least one of the factors that will help. There are so many factors something working out is likely.
And with practice:
3) $12 = 13 -1$ and $13$ is prime. So Euler's Theorem/Fermats Little Theorem will probably come into play.
.....
Factor $n^{12} -a^{12}$ completely.
$n^{12}-a^{12} = (n-a)(n^11 + n^10a + .... +na^{11} + a^{12})=$
$(n^2 - a^2)(n^{10} + n^8a^2 + .... + n^2a^8 + a^{10})=$
$(n^3 - a^3)(n^9 + n^6a^3 + n^3a^6 + a^9)=$
$(n^4-a^4)(n^8 + n^4a^4 + a^8)=$
$(n^6 - a^6)(n^6 + a^6) =$
$(n^6+a^6)(n^3 + a^3)(n^2+a^2)(n+a)(n-a)$.
Those probably have different remainders when divided by $7$ or $13$. One of them can probably be proven to be $0$. but... that may be hard to figure out. Let's put a pin in it and look at 3)
3) Fermat's little theorem states if $\gcd(k,13)=1$ then $k^{12}\equiv 1 \pmod {13}$. And $n,a$ are coprime to $91$ so they are coprime to $13$ so
$a^{12} \equiv 1 \pmod {13}$ and $n^{12} \equiv 1 \pmod {13}$ so $n^{12}-a^{12}\equiv 1-1 \equiv 0 \pmod 13$ so $13|(n^{12} - a^{12})$.
So now we just need to prove $7|n^{12} - a^{12}$....
Back to the factoring... oh, wait!....
Fermat's Little Theorem states $n^6\equiv 1\pmod 7$ and $a^6\equiv \pmod 7$ so $n^{12}-a^{12} = (n^6- a^6)(n^6 + a^6)\equiv (1-1)(n^6 + a^6)\equiv 0 \pmod 7$ so $7|n^{12}-a^{12}$.
And we are done!
We didn't need to do that factoring stuff at all.
Still it'd be interesting to see if we can do it by considering just the factors. But we don't have to.
