Each of the numbers $a$ and $b$ can be expressed as the product of three prime numbers less than 10. Each prime number less than 10 is present in the factorization of at least one of the numbers $a$ and $b$. The greatest common divisor of the numbers $a$ and $b$ is equal to the greatest common divisor of the numbers $\frac{a}{15}$ and $b$ and simultaneously twice the greatest common divisor of the numbers $a$ and $\frac{b}{4}$.

Determine the numbers $a$ and $b$.

(E. Semerádová)
Prime numbers less than 10 are $2, 3, 5$, and 7. From the problem statement, it follows that the number $a$ is divisible by 15 and the number $b$ is divisible by 4. The prime factorizations of these numbers are therefore of the form

$$
a=3 \cdot 5 \cdot p, \quad b=2 \cdot 2 \cdot q
$$

where $p$ and $q$ are prime numbers from the list above.

Since the greatest common divisor of the numbers $a$ and $b$ is equal to twice the greatest common divisor of the numbers $a$ and $\frac{b}{4}$, it must be that $p=2$ and $q \neq 2$. Thus,

$$
a=3 \cdot 5 \cdot 2=30 \text{. }
$$

Since the greatest common divisor of the numbers $a$ and $b$ is equal to the greatest common divisor of the numbers $\frac{a}{15}$ and $b$, $q$ cannot be 3 or 5. From the previous, we know that $q$ cannot be 2 either. Thus, $q=7$ and

$$
b=2 \cdot 2 \cdot 7=28
$$

Each of the prime numbers listed is present in the factorization of at least one of the numbers $a$ and $b$. The unknown numbers are $a=30$ and $b=28$.

Evaluation. 1 point for the divisibility of the number $a$ by fifteen; 1 point for the divisibility of the number $b$ by four; 3 points for the discussion of the possibilities for $p$ and $q$; 1 point for determining the correct possibility and expressing $a$ and $b$.