3. (3 points) Anya, Vanya, Danya, Sanya, and Tanya were collecting apples. It turned out that each of them collected a whole percentage of the total number of collected apples, and all these numbers were different and greater than zero. Then Tanya, who collected the most apples, ate her apples. After that, it turned out that each of the children still had a whole percentage, but now of the remaining number of apples. What is the minimum number of apples that could have been collected?
Answer: 20, for example $1+2+3+4+10$

## Solution:

Let Anna have collected the fewest apples, and this number is $k$, and the total number of apples is $n$. Then we get that the total number of apples collected is no less than $4 k+10$. For $k \geq 3$, we already get more than 20 apples. For $k=2$, we get $n \geq 18=9 k$. Since $\frac{n}{k}$ must be an integer and a divisor of a hundred, it turns out that no fewer than $10 k$ apples were collected, that is, at least 20 again. For $k=1$, we get that the children collected at least 15 apples, and this number, as in the previous considerations, must be a divisor of a hundred, that is, at least 20 apples again.