Prove that:
6n−5n+4 is divisible by 5 for n≥1
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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