For how many natural numbers(<=100) is $1111^n +2222^n+3333^n+4444^n$ divisible by 10? For how many natural numbers (0 not included) $n \leq 100$ is $1111^n +2222^n+3333^n+4444^n$ divisible by 10?
I factored out $1111^n$ and got $1111^n(1+2^n+3^n+4^n)$. So $1+2^n+3^n+4^n$ must be divisible by 10. I figured out that this is divisible by 10 for all odd n, but I don't know how to find the other solutions, if any.

$$1^n+2^n+3^n+4^n\equiv0\pmod2\text{ for }n\ge0$$
As $2^3\equiv3,2^2\equiv4\pmod5,$
$$1^n+2^n+3^n+4^n$$
$$\equiv1+2^n+(2^3)^n+(2^2)^n$$
$$\equiv\begin{cases}\dfrac{2^{4n}-1}{2^n-1}\equiv 0 &\mbox{if } 2^n-1\not\equiv0\pmod5\iff4\nmid n  \\
4 & \mbox{if } 4\mid n \end{cases}\pmod5$$
$$\implies1^n+2^n+3^n+4^n\equiv0\pmod{[5,2]}\text{ if }4\nmid n$$
