The last two digits of a three-digit number are equal. The number equals the sum of thrice the product of its digits and ten times the sum of its digits. How many such three-digit numbers are there?

To solve this problem, we need to understand the conditions given and translate them into mathematical equations.

1. The last two digits of a three-digit number are equal. This means our number looks like this: XYX, where X and Y are digits from 0 to 9.

2. The number equals the sum of thrice the product of its digits and ten times the sum of its digits. This can be translated into the following equation: 100X + 10Y + X = 3 * X * Y * X + 10 * (X + Y + X).

Simplify the equation: 101X + 10Y = 3X^2Y + 20X + 10Y.

Further simplification gives: 81X = 3X^2Y.

Now, we can start testing possible values for X and Y. Since X and Y are digits, they can only be integers from 0 to 9. However, since we are looking for a three-digit number, X cannot be 0. So, X can be from 1 to 9, and Y can be from 0 to 9.

By testing these values, we find that the only solutions that satisfy the equation are X = 3, Y = 0 and X = 3, Y = 9. So, the three-digit numbers that satisfy the conditions are 303 and 393.

Therefore, there are 2 such three-digit numbers.