# Problem 2. (3 points)

Natural numbers $a$ and $b$ are such that $2a + 3b = \operatorname{LCM}(a, b)$. What values can the number $\frac{\operatorname{LCM}(a, b)}{a}$ take? List all possible options in ascending or descending order, separated by commas. If there are no solutions, write the number 0.
Answer: 0 (no solutions)

## Solution:

Let $d=\operatorname{GCD}(a, b)$. Then $a=x d, b=y d, \operatorname{LCM}(a, b)=x y d$, where $x$ and $y$ are coprime. We get $2 x d+3 y d=x y d$. Dividing by $d$ yields $2 x+3 y=x y$.

The right side of this equation is divisible by $x$, so $2 x+3 y$ is divisible by $x$, which means $3 y$ is divisible by $x$. Since $x$ and $y$ are coprime, 3 must be divisible by $x$. Therefore, $x=3$ or $x=1$. Similarly, 2 must be divisible by $y$.

If $x=3$, the equation becomes $6+3 y=3 y$, which is false. If $x=1$, we get $2+3 y=y$, from which $y$ is negative, which is also impossible.

Therefore, there are no solutions.