The product of two two-digit numbers is 1728. If the greatest common divisor $(gcd)$ of them is 12, what are these numbers?
As 12 is the greatest common divisor of the two numbers, both are multiples of 12, so they are among the numbers

$$
12,24,36,48,60,72,84,96,108,120,132,144, \ldots
$$

From the list above, we have three unique possibilities:

$$
\begin{aligned}
& 12 \times 144=1728 \text { and } \gcd(12,144)=12 \\
& 24 \times 72=1728 \quad \text { and } \quad \gcd(24,72)=24 \\
& 36 \times 48=1728 \quad \text { and } \quad \gcd(36,48)=12
\end{aligned}
$$

Therefore, we have two solutions: 12 and 144, or 36 and 48.

$$
\begin{aligned}
& 12=2^{2} \times 3,144=2^{4} \times 3^{2} \\
& \gcd(12,144)=2^{2} \times 3
\end{aligned}
$$

72 is a multiple of 24, $\Rightarrow \gcd(24,72)=24$

$36=2^{2} \times 3^{2}$ and $48=2^{4} \times 3$, $\Rightarrow \gcd(36,48)=2^{2} \times 3$