2.1. Solve the equation $\operatorname{GCD}(a, b)+\operatorname{LCM}(a, b)=a+b+2$ in natural numbers.
2.1. Solution. We will use standard notation: $(a, b)=\operatorname{GCD}(a, b)$ and $[a, b]=\operatorname{LCM}(a, b)$. Notice that from the condition it follows that 2 divides $(a, b)$. Therefore, we have two cases: $(a, b)=1$ and $(a, b)=2$. In the first case, we get that $[a, b]=a b$ and we have the equation $a b+1=a+b+2$ or $(a-1)(b-1)=2$. From this, we get the solutions $(3,2)$ and $(2,3)$. If $(a, b)=2$, then $[a, b]=a b / 2$, and then we have the equation $\frac{a b}{2}+2=a+b+2$, from which $(a-2)(b-2)=4$. Such an equation has solutions $(6,3)$ and $(4,4)$, but none of them fit, as in both cases $(a, b) \neq 2$.