3. In the box, there are balls of seven colors. One tenth of the balls are red, one eighth are orange, and one third are yellow. There are 9 more green balls than red ones, and 10 more blue balls than orange ones. There are 8 blue balls in the box. The remaining balls are purple. What is the smallest possible number of purple balls?
Answer: 25 balls.

Solution. Let the total number of balls be $x$, and the number of violet balls be $y$. Then

$$
\frac{x}{10}+\frac{x}{8}+\frac{x}{3}+\frac{x}{10}+9+\frac{x}{8}+10+8+y=x
$$

from which $\frac{47 x}{60}+27+y=x$, that is, $y=\frac{13 x}{60}-27$, and the smallest value of $y$ is achieved at the smallest value of $x$. Since $y$ is an integer, $x$ must be a multiple of 60. According to the condition, $x$ must also be a multiple of 8, so $x$ is divisible by 120. For $x=120, y=-1$, which is impossible. For $x=240, y=25$, and this is the smallest possible number of violet balls.

Comment. The correct solution and the correct answer are given - 7 points. The correct answer is found by trial and error, and the minimality is not proven - 3 points. If the correct answer is not obtained, but the idea of divisibility is considered in the solution, which is implemented incorrectly - 1 point. For arithmetic errors with correct reasoning, deduct 1-3 points.