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For how many positive integers n less than 100 is 5^n+8^{n+1}+11^{n+2}+5^{n+2} a multiple of 6?

 Oct 25, 2022
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0 < n < 100

\(5^n + 8^{n + 1} + 11^{n + 2} + 5^{n+2} \)

and the expression above is divisible by 6.

5^n will always end in 5.

11^m will always end in 1.

That means 5^n + 5^(n + 2) will end in 0.

8^(n + 1) will end in an even number, meaning that the expression will be odd because of 11^(n+2). 

If a number has to be even to be divisible by 6, then that means:

The expression will never be divisible by 6 for 0 < n < 100.

 Oct 25, 2022
edited by proyaop  Oct 25, 2022

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