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?
+1622 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.
