If we know the GCD and LCM, how to find all unordered pairs? If we know that $\gcd(a,b)=12$ and $\operatorname{lcm}(a,b)=360$, how can we find all unordered pairs $a,b$?
The answer in the back of the book I got this from gives $(12,360), (24,180), (36,120), (60,72)$ as the solutions.

If $\gcd(a,b) = 12$, then $12 \mid a$ and $12 \mid b$. Hence $\dfrac{a}{12}$ and $\dfrac{b}{12}$ are integers and 
$\gcd\left( \dfrac{a}{12}, \dfrac{b}{12} \right) = 1$
. Hence...
\begin{align}
   ab &= \gcd(a,b) \times  \operatorname{lcm}(a,b) \\
   ab &= 12 \times 360 \\
   \dfrac{a}{12} \dfrac{b}{12} &= \dfrac{360}{12} \\
   \dfrac{a}{12} \dfrac{b}{12} &= 30 \\
\end{align}
So, if you are looking for unordered pairs,
  $\left(\dfrac{a}{12}, \dfrac{b}{12} \right) 
   \in \{ (1,30), (2,15), (3,10), (5,6)\}$
Note that the condition $\gcd\left( \dfrac{a}{12}, \dfrac{b}{12} \right) = 1$ has been satisfied in each case.
So,
$(a,b) \in \{ (12,360), (24,180), (36,120), (60, 72)\}$
