15. Find a three-digit number that is equal to the cube of the sum of its digits.
15. Since $4^{3}=64$ and $10^{3}=1000$, the sum of the digits can be $5,6,7,8$ or 9. From these five numbers, we will select by trial those that satisfy the condition of the problem:

$$
5^{3}=125,6^{3}=216,7^{3}=343,8^{3}=512,9^{3}=729
$$

The condition is satisfied by $512=(5+1+2)^{3}$.