For a number to have exactly three factors, it must be a perfect square.
So, each of x, y, and z must be a perfect square.
Let x = a^2, y = b^2, and z = c^2, where a, b, and c are distinct natural numbers.
Then, x^2y^3z^4 = (a^2)^2(b^2)^3(c^2)^4 = a^4b^6c^8.
The number of factors of a^4b^6c^8 is (4+1)(6+1)(8+1) = 5*7*9 = 315.
Therefore, x^2y^3z^4 has 315 factors.
The answer is: 315
Below is a math problem. Provide a detailed, step-by-step solution.

### Problem:
If three distinct natural numbers, x, y, and z, each have exactly three factors, how many factors does x^2y^3z^4 have?

### Answer:
