Show that $37|333^{777} + 777^{333}$

Question

Show that $37|333^{777} + 777^{333}$. Anyone can help me to solve this question? I have no idea how to solve it.

Answer 1

$37*3 =111$ so $37$ divides any linear combination of $111$.

And $333^{777} + 777^{333} = 111^{77}*3^{777} + 111^{333}*7^{333}$ which is clearly divisible by $111$ and therefore by $37$.

In particular $333\div 37 = 9$ so $333^{777} = 333*333^{776} = 37*9*333^{776}$ and $777\div 37 = 21$ so $777^{333} = 37*21*777^{332}$ and so $333^{777}+ 777^{333} = 37(9*333^{776} + 21*777^{332})$.

Answer 2

The hint:

We have: $$111=37\cdot3.$$