I have tried understanding how to solve questions of these type using pen and paper, without access to a calculator.
Here's the question:
What is the remainder of $2019^2 + 2019^4 + 2019^6 + 2019^8$ $mod 4$ ?
I know that there's a pattern to these types of questions, but I am not exactly sure how to find it.
I thought about breaking the numbers into smaller numbers that are congruent to them, but I am not sure if that would be of any help.
Any help would be greatly appreciated!
Since $\;2019=-1\pmod 4\;$ ,we get
$$2019^2+2019^4+2019^6+2019^8=(-1)^2+(-1)^4+(-1)^6+(-1)^9=1+1+1+1=0\pmod4$$