<s>[INST] 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 $$n\neq 0,4,8,12,16.....$$

 [/INST] HINT : You can consider each right-most digit.
$$1:1,1,1,1,1,\cdots$$
$$2^n:2,4,8,6,2,\cdots$$
$$3^n:3,9,7,1,3,\cdots$$
$$4^n:4,6,4,6,4,\cdots$$
So, we have
$$1+2+3+4=1\color{red}{0},1+4+9+6=2\color{red}{0},1+8+7+4=2\color{red}{0},1+6+1+6=14.$$
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