499. Find all pairs of natural numbers whose greatest common divisor is 24, and the least common multiple is 360.
$\triangle$ Let the required numbers be $a$ and $b$.

We factorize the numbers 24 and 360 into prime factors:

$$
24=2^{3} \cdot 3, \quad 360=2^{3} \cdot 3^{2} \cdot 5
$$

Therefore, the prime factorizations of the numbers $a$ and $b$ contain only twos, threes, and fives, and $2^{3}=8$ is a factor of each of the numbers $a$ and $b$, $3$ is a factor of one of the numbers in the second power and of the other in the first power, and $5$ is a factor of one of these numbers. Let's consider all possibilities.

1) Suppose

$$
a=2^{3} \cdot 3=24, \quad b=2^{3} \cdot 3^{2} \cdot 5=360
$$

We get the numbers 24 and 360.

2) Suppose

$$
a=2^{3} \cdot 3^{2}=72, \quad b=2^{3} \cdot 3 \cdot 5=120
$$

We have found the numbers 72 and 120.

Obviously, these are all the cases.

Answer: 24 and $360 ; 72$ and 120.