ช่วยหน่อย ให้ a และ b เป็นจำนวนเต็มซึ่ง (a,a+3b) = 5 และ ab = 110 จงหา [a,3b]

[optimus]

ให้ a และ b เป็นจำนวนเต็มซึ่ง (a,a+3b) = 5 และ ab = 110 จงหา [a,3b]

[Anton]

Problem. Two integers $a$ and $b$ satisfy $$(a,a+3b)=5$$ and $$ab=110\,.$$ Find the value of $[a,3b]$.

Presumably, $(\_,\_)$ and $[\_,\_]$ denote the greatest common divisor ($\gcd$) function and the least common multiple ($\operatorname{lcm}$) function, respectively.
Because $\gcd(a,a+3b)=\gcd\big(a,(a+3b)-a\big)=\gcd(a,3b)$, we conclude that $\gcd(a,3b)=5$. We also note that
$$3ab=3\cdot 110=330\,.$$
Hence,
$$\operatorname{lcm}(a,3b)\cdot \gcd(a,3b)=|a|\,|3b|=|3ab|=330$$
implies that
$$\operatorname{lcm}(a,3b)=\frac{330}{\gcd(a,3b)}=\frac{330}{5}=66\,.$$

It should however be seen that the hypothesis of this problem is impossible to satisfy. This is because
$$\gcd(x,y)\mid\operatorname{lcm}(x,y)$$
for all integers $x$ and $y$, but $5\nmid 66$. This is either a trick question or a mistake by the problem designer, or quite possibly, a mistake by the asker (optimus). If it is a mistake by the problem designer, then the problem designer aimed for the answer to be $66$.

Alternatively, one can already see that $\big(\gcd(x,y)\big)^2$ is a factor of $xy$ for any integers $x$ and $y$. For the problem at hand, $5^2$ does not divide $330$.