<s>[INST] Number Theory divisibilty How can I check if $$12^{2013} + 7^{2013}$$ is divisible by $19$?
Also, how can I format my questions to allow for squares instead of doing the ^ symbol. 
 [/INST] First we prove two theorems that we will use

If $a,\;b,\;c\;\in \mathbb{N}$ with $a\neq0$ and $x,y\in\mathbb{N} $such that $a\mid b$ and $a\mid c$, then, $a\mid bx+cy$;

Show:$a\mid b$ and $a\mid c$ implies that there $m,n\in\mathbb{N}$ such that $b=a\cdot m$ and $c=a\cdot n$;$$bx+cy=am\cdot x+an\cdot y=a(mx+ny)\Longrightarrow a\mid bx+cy \;\;\;\Box$$Now we have to prove the theorem we use to solve this issue, and in this test we use the theorem proved above

Are $a,b,n\in\mathbb{N}$ with $a+b\neq0$ we have $a+b\mid a^{2n+1}+b^{2n+1}$ 

Show: Have demonstrated induction $n=0$ the statement is true because $a+b\mid a^{2\cdot 0+1}+b^{2\cdot0+1}\Longrightarrow a+b\mid a+b$; hypothesis: $a+b\mid a^{2n+1}+b^{2n+1}$;$$a^{2(n+1)+1}+b^{2(n+1)+1}=a^{2n+3}+b^{2n+3}=a^{2n+1+2}+b^{2n+1+2}=a^2a^{2n+1}+b^2b^{2n+1}=$$$$=a^2a^{2n+1}+b^2b^{2n+1}\underbrace{-b^2a^{2n+1}+b^2a^{2n+1}}_{=0}=a^2a^{2n+1}-b^2a^{2n+1}+b^2b^{2n+1}+b^2a^{2n+1}=$$$$=a^{2n+1}(a^2-b^2)+b^2(a^{2n+1}+b^{2n+1})$$we know that $a+b\mid a^2-b^2$ because $a^2-b^2=(a+b)(a-b)$; our hypothesis gives us that $a+b\mid a^{2n+1}+b^{2n+1}$ and the theorem shown above assures us that $a+b\mid a^{2n+1}(a^2-b^2)+b^2(a^{2n+1}+b^{2n+1})$ thus showing that the theorem is valid for $n +1$, then, goes for any $n\in\mathbb{N}\;\;\Box$

Question: Show that $19\mid 12^{2013}+7^{2013}$

Using the theorem proved above, we have easily that $19=12+7$ and $$12+7\mid 12^{2013}+7^{2013}$$because $2013=2\cdot1006+1$ and $1006\in\mathbb{N}$
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