1. Find the smallest $n \geq 2017$, such that $1^{\mathrm{n}}+2^{\mathrm{n}}+3^{\mathrm{n}}+4^{\mathrm{n}}$ is not divisible by 10.
Answer: 2020.

Solution: Powers of 1 always end in 1. The last digit of powers of 2 changes with a period of 4:

![](https://cdn.mathpix.com/cropped/2024_05_06_35052e7d512455511f98g-07.jpg?height=362&width=688&top_left_y=1121&top_left_x=1209)

$2,4,8,6$. Powers of 3 also follow the pattern $3,9,7,1$. Powers of 4 change with a period of 2: 4,6,4,6. Therefore, the last digit will repeat with a period of 4. Checking for $n=1,2,3$, we find that the last digit is 0, and for $n=4$ it is 4. Therefore, the smallest N>2016 will be 2020.