<s>[INST] For what $n$ does $x^n \equiv 2\pmod{13}$ have a solution? I want to ask for what values of $n$ the congruence $$x^n \equiv 2 \pmod{13}$$ has a solution for $x$.
 [/INST] Hints: Note that $x^{12k+m}\equiv x^m\pmod{13}$. So you will "only" have to examine the cases $m=0$ to $11$. 
If $m$ and $12$ are relatively prime, then the congruence $x^m\equiv a\pmod{13}$ always has a solution. 
So we are down to $m=0$, $2$, $3$, $4$, $6$, $8$, $9$, $10$. 
You can find shortcuts to deal with some of these. For example, once you find that $x^2\equiv 2$ has no solutions, that rules out the even $m$ in the list. (One can use a theorem to rule out $m=2$, or one can square the numbers $1$ to $6$.) There are other theorems one could quote to bypass calculations.
Remark: The following theorem lets us deal quickly with the problem,with minimal calculation. 
Theorem:  Let $a$ be relatively prime to the prime $p$. Then the congruence $x^k\equiv a\pmod{p}$ has a solution if and only if $a^{(p-1)/d}\equiv 1\pmod{p}$, where $d=\gcd(k,p-1)$.
Let $p=13$ and $a=2$. The possible $d$ are $1,2,3,4,6,12$. Since $2^4\not\equiv 1\pmod{13}$ and $2^6\not\equiv 1\pmod{13}$, it follows that $x^k\equiv 2\pmod{13}$ has a solution if and only if $\gcd(k,12)=1$.  
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