Question

What is the remainder of 2019 to the 2019th power when divided by 2020?​

Answer 1

a good question after a long time....

we're asked

`{2019}^(2019) \%2020`

we can write,

`{2019}^(2019) = {(2020 - 1)}^(2019)`

now apply binomial expansion,

`\binom{2019}{0} {2020}^(2019) {( - 1)}^(0) + \binom{2019}{1} {2020}^(2018) {( - 1)}^(1) + \binom{2019}{2} {2020}^(2017) {( - 1)}^(2) ... \binom{2019}{1} {2020}^(1) {( - 1)}^(2018) + \binom{2019}{2019} {2020}^(0) {( - 1)}^(2019)`

if you notice, each term except Last has 2020 in it, which would get divided... so the remainder is 1