Thursday, 1 September 2016

elementary number theory - Proof of divisibility using modular arithmetic: 5mid6n5n+4



Prove that:



6n5n+4 is divisible by 5 for n1



Using Modular arithmetic. Please do not refer to other SE questions, there was one already posted but it was using induction, I want to use this number theory method.




Obviously we have to take \pmod 5



So:



6^n - 5n + 4 \equiv x \pmod 5



All we need to do prove is prove x = 0



How do we do that? I just need a hint, I am not sure how to solve congruences. Some ideas will be helpful.




Thanks!


Answer



Hint:-
6\equiv1 \pmod 5\implies 6^n\equiv1\pmod 5\tag{1}



-5(n-1)\equiv 0\pmod 5\tag{2}



Solution:-





(1)+(2) gives,6^n-5n+4\equiv0\pmod 5



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