Proving $1+2^n+3^n+4^n$ is divisible by $10$ How can I prove 

$$1+2^n+3^n+4^n$$ is divisible by $10$  if $$n\neq 0,4,8,12,16.....$$


$$3\equiv1\pmod2\implies 1+2^n+3^n+4^n\equiv0\pmod2$$
If $n$ is odd, $1+4^n=1^n-(-4)^n$ is divisible by $1-(-4)=5$
Similarly, $2^n+3^n\equiv0\pmod5,$
For $n=4k+2,2^{4k+2}\equiv4\pmod5,3^{4k+2}\equiv-1,4^{4k+2}\equiv1$
