The number of ways that  $2^n+3^n+5^n$ is divisible by 4 where  $n\in \{1, 2, 3......, 100\}$ is

If n is odd, then $3^n+5^n$ is divisible by $(3+5)$
$n = 3,5,.......99$  which are 49 ways
If n is even then  $2^n+3^n+5^n = 4k_1+4k_2+1+4k_3+1 = 4k+2$
which is not divisible by 4