2. (2 points) Boys were collecting apples. Each collected either 10 apples or $10\%$ of the total number of apples collected, and there were both types. What is the smallest number of boys there could have been?
# Answer: 6

Solution: Example: one boy collected 10 apples, the rest collected 2 each, totaling 20.

Evaluation: Let's prove that there could not have been fewer boys. We know that $10\%$ of the total number of apples is an integer, let's denote it as $k$. Suppose $n$ people collected ten apples each, and $m$ people collected $10\%$ each, then $10n = (10-m)k$, which means $10n$ is divisible by $10-m$. When $m=1$ or $m=3$, we get that $10n$ is divisible by 9 or 7, respectively, meaning $n$ is divisible by 9 or 7, which is too large. When $m=2$ or $m=4$, we get that $10n$ is divisible by 8 or 6, respectively, meaning $n$ is divisible by 4 or 3, and $m+n \geqslant 6$. When $m \geqslant 5$, the total number of boys is at least 1 more, meaning, again, at least 6.