Problem 1. Vasya multiplied a certain number by 10 and got a prime number. And Petya multiplied the same number by 15, but still got a prime number. Could it be that neither of them made a mistake?

[3 points]
Answer: Yes.

Solution: The number could have been 0.2. Indeed, $0.2 \cdot 10 = 2$ and $0.2 \cdot 15 = 3$ - both are prime numbers.

Comment: It can be proven that the given example is the only one. Let Vasya get the number $p$, and Petya get the number $q$. According to the problem, $\frac{p}{10} = \frac{q}{15}$. Therefore, $3p = 2q$. This means that $p$ is an even prime number, i.e., $p = 2, q = 3$, and the original number $\frac{p}{10} = \frac{q}{15} = 0.2$.