For how many positive integers $n$ less than $100$ is $5^n+8^{n+1}+13^{n+2}$ a multiple of $6$?
+289 5^n + 8^(n+1) + 13^(n + 2) = (-1)^n + 2^(n+1) + 1 (mod 6)
Case 1: n is odd
2^(n + 1) = 0 (mod 6)
No values of n will satisfy this as we will need to get a factor of 3 in order to make this expression divisible by 6.
Case 2: n is even
2^(n+1) + 2 = 0 (mod 6)
2^(n+1) = 4 (mod 6)
2
4 check
8
16 check
32
64 check
128
256 check
It seems like all the even powers of 2 leave a remainder of 4 when divided by 6.
However, if we subtract one, we would get an odd number for n which goes against the case restrictions. Therefore, there are 0?
Please correct me if I am wrong.
