TITLE: For all positive integers $n$, $14^{6n} - 11^{6n}$ is divisible by?
QUESTION [5 upvotes]: For all positive integers $n$, $14^{6n} - 11^{6n}$ is divisible by ?

This question is followed with four options: 
$1)157\quad\quad
  2) 163\quad\quad
  3) 225\quad\quad
  \quad 4) \text{All of these}$
I don't know how to do this in a "fast" way manually,what I did is used computer and factorized $14^6-11^6$ which gives $5757975=157\times 163 \times 225$ but surely this is not what I am supposed to do during exams as it will be too tedious even if I check  for divisibility of the three numbers manually using the options but does computing $14^6-11^6$  and then checking for divisibility is  the best option for solving this problem (manually/using pencil and paper)?

REPLY [8 votes]: HINT $\rm\ x^6 - y^6\ =\ (x-y)\ (x+y)\ (x^2+y^2-xy)\ (x^2+y^2+xy)$   
$\qquad\qquad\qquad\qquad\qquad =\ 3\cdot 5^2\ (317-154)\ (317+154)\ =\ 3\cdot 5^2\cdot 163\ ( 3\cdot 157)\ \ $ for $\rm\:x,y = 14,11$  
Since $\ 3^2\cdot 5^2\: =\: 15^2\: =\: 225\:,\ $ we deduce $\rm\:157,\:163,\: 225\ |\ 14^6-11^6\ |\ 14^{\:6\:n}-11^{\:6\:n}\:.$