10.1. Every second, the computer outputs a number equal to the sum of the digits of the previous number, multiplied by 31. On the first second, the number 2020 was displayed. What number will be displayed on the 2020th second
Answer: 310.

Solution: Let's calculate the first few numbers displayed on the screen.

On the first second, the number $a_{1}=2020$ is displayed, then

on the second second, the number $a_{2}=(2+0+2+0) \cdot 31=4 \cdot 31=124$ is displayed;

on the third second, the number $a_{3}=(1+2+4) \cdot 31=7 \cdot 31=217$;

on the 4th second, the number $a_{4}=(2+1+7) \cdot 31=310$;

on the 5th second, the number $a_{5}=(3+1+0) \cdot 31=4 \cdot 31=124=a_{2}$.

Since each subsequent number is calculated using only the previous number, the numbers displayed on the screen will repeat with a period of 3. The number of seconds (2020 - 1) is divisible by 3, so on the 2020th second, the number 310 will be displayed.

# Criteria: 

7 points - correct answer and complete justification;

5-6 points - correct answer and generally correct reasoning, with minor gaps or inaccuracies

4 points - correct line of reasoning provided, but a computational error was made;

answer only - 0 points.