Prove that: $\displaystyle5^{2012}+1$ is divisible with $313$.
What I try and what I know:
$313$ is prime and I try use the following formula :
$a^n+b^n=(a+b)(a^{n-1}-a^{n-2}b+\ldots\pm(-1)^{n}b^{n-1})$
but still nothing. this problem can be solved using a elementary proof because I found it a mathematical magazine for children with the age of 14.