Sorry for the english translation.
$x$ and $y \in \Bbb R$. $15$ is the minumum integer which makes $x$ integer when you multiply. $18$ is the minumum integer which makes $y$ an integer when you multiply with. What is the minumum factor which makes $4x+2y$ integer?
How can I start? İsn't it $45|(4x+1)$?
Write $x=\frac a{15}$ and $y=\frac b{18}$. These fractions are in their lowest terms. Then $$4x+2y=\frac{4a}{15}+\frac b9=\frac{12a+5b}{45}$$ So we know that $45(4x+2y)$ is an integer. Furthermore, this latter fraction is also in their lowest term. Let's show it:
The prime factors of $45$ are $3$ and $5$. Since $\gcd(b,18)=1$, then $b$ is not a multiple of $3$ and $12a+5b$ is not either. Similarly, since $\gcd(a,15)=1$, $a$ is not a multiple of $5$ and then $12a+5b$ is not either.
We conclude that $45$ is the least integer $k$ that makes $k(4x+2y)$ an integer.
