My favorite numbers are those which, when multiplied by their own digit sum, become ten times larger. The product of three of my favorite numbers is 71668. Which are these numbers?

(Birdie)
A favorite number is one that, when multiplied by the sum of its digits, increases tenfold. This means that the sum of the digits of a favorite number must be 10. For example, the number 73 has a digit sum of 10, so $73 \cdot 10 = 730$. Let's factorize the number 71668 into prime factors:

$$
71668 = 2 \cdot 2 \cdot 19 \cdot 23 \cdot 41
$$

(2 points for the factorization)

Among these, only the number 19 has a digit sum of 10. The numbers 2, 23, and 41 are not favorite numbers, but by multiplying them appropriately, we can obtain numbers with a digit sum of 10. Therefore, we multiply $2 \cdot 23 = 46$ and $2 \cdot 41 = 82$, which gives us the desired numbers.

Thus, I multiplied the numbers 19, 46, and 82.

(6 points for the complete solution)