For how many positive integers $n$ less than $100$ is $5^n+8^{n+1}+13^{n+2}$ a multiple of $6$?

Guest Oct 8, 2022

#2**0 **

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.

Voldemort Oct 9, 2022