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

Václav Mordvinov
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Ka Shing Ma
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    Hint: are $333$ and $777$ divisible by $37$? – Mark Oct 22 '18 at 20:36
  • Welcome to MathStackExchange. I would suggest you edit your question to provide some additional detail to help us help you. What is the context of this question? Is it a homework question? What kind of course are you taking? It is also good practice to show some work, either explaning where your difficulties are or writing down your attempt to solve the problem. – Mefitico Oct 22 '18 at 20:37
  • Duplicate of this – Bill Dubuque Oct 22 '18 at 20:50
  • I am taking elementary number theory course and this question is one of my homework question. But I have no idea how to solve this kind of question, hope that I can find the answer or hint form here. Thank you so much. – Ka Shing Ma Oct 22 '18 at 20:51
  • Well, for sure you tried something? When you ask a question it is always better to write what about your attempt to solve it. Anyway, the hint I gave you should make it very easy. – Mark Oct 22 '18 at 20:54
  • "But I have no idea how to solve this kind of question, " Really? You could start be trying to divide $37$ into $333^{777} + 777^{333}$. It wouldn't take much to find $333\div 37 = 9$ and $777\div 37 = 21$ so $(333^{777}+ 777^{333} = 37(9333^{776} + 21777^{332})$. Admittedly, it seems like it wouldn't be that simple.... but it was. – fleablood Oct 22 '18 at 21:18

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The hint:

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

Michael Rozenberg
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$37*3 =111$ 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 $333^{777}+ 777^{333} = 37(9*333^{776} + 21*777^{332})$.

fleablood
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