TITLE: Can this number theory MCQ be solved in 4 minutes?
QUESTION [7 upvotes]: The Problem: ( 270 + 370 ) is divisible by which number? [ 5, 13, 11 , 7 ]

Using Fermat's little theorem it took more than the double of the indicated time limit. But I would like to solve it quickly as I am preparing for an exam, that would require us to solve each MCQs to be solved in 4 minutes on an average. 

Thanks in advance.

REPLY [3 votes]: Besides noticing $\ 2^2+3^2\ |\ 2^{70}+3^{70}\ $ here is another quick way. $\:$ Mod $\rm\:m = 5,7,13\:$ we have $\rm\: x\not\equiv 0\ \Rightarrow\ x^{12} \equiv 1\ \Rightarrow\ x^{70}\equiv x^{-2}\:.\ $ So $\rm\ 2^{70} + 3^{70}\equiv 1/4 + 1/9 \equiv 13/36\equiv 0\iff m = 13\:.$