10.1. The first 20 numbers of the natural number sequence were written on the board. When one of the numbers was erased, it turned out that among the remaining numbers, one is the arithmetic mean of all the others. Find all the numbers that could have been erased.
Answer: 1 and 20.

Solution. The sum of the first twenty numbers in the natural number sequence is $(1+20) \cdot 10=210$. Therefore, if one of them is erased, the sum $S$ of the remaining numbers satisfies the inequality $190 \leq S \leq 209$. The arithmetic mean of the remaining numbers is $S / 19$. According to the condition, this is a natural number, so $S$ is a multiple of 19. In the interval $[190,209]$, there are exactly two such numbers: 190 and 209. If $S=190$, then the number 20 was erased, and if $S=209$, then the number one was erased.