5. (3 points) Solve the equation $a+b+\operatorname{GCD}(a, b)+\operatorname{LCM}(a, b)=143$.
Answer: No solutions

Solution:

Let $d=\text{GCD}(a, b)$. Then for some $k$ and $l$, the equations $a=k d, b=l d, \text{LCM}(a, b)=k l d$ hold. Accordingly, the equation becomes $k d+l d+d+k l d=143$ or $(k+1)(l+1) d=11 \cdot 13$. Since $k+1$ and $l+1$ are both greater than 1, one of them must be 11 and the other 13. Therefore, $d=1, a=k, b=l$, which means there are two possible answers: $a=10, b=12$ or vice versa.

However, these numbers are not coprime ($d \neq 1$), so they actually do not fit.