The product of two two-digit numbers is 1728. If their greatest common divisor (GCD) is 12, what are these numbers?
As 12 is the GCD of the two numbers and each has two digits, the only candidates are the multiples of 12 less than 100, that is,

$$
12,24,36,48,60,72,84 \text { and } 96
$$

Since $1728=12 \times 12 \times 12=2^{6} \times 3^{3}$, the multiples 60 (with factor 5) and 84 (with factor 7) are not divisors of 1728. Also, $1728 \div 12=144$ and $1728 \div 96=18$, so the list reduces to 24, 36, 48, and 72, with $24 \times 72=36 \times 48=1728$. Since the GCD of 24 and 72 is 24, we have a unique solution, namely 36 and 48, whose product is 1728 and the GCD is 12.