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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!

Abwatts
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Since $\;2019=-1\pmod 4\;$ ,we get

$$2019^2+2019^4+2019^6+2019^8=(-1)^2+(-1)^4+(-1)^6+(-1)^9=1+1+1+1=0\pmod4$$

DonAntonio
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  • Thanks a lot! So it's basically all about trial and error until we find what 2019 is congruent to? – Abwatts Dec 13 '19 at 21:58
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    Well, no: it is just basic arithmetic...and the fact that $;4;$ is a pretty small number and it is very easy to know what residue modulo $;4;$ any integer is. – DonAntonio Dec 13 '19 at 22:26