There are 5 odd digits: 1, 3, 5, 7, and 9.
For the hundreds digit, we have 5 choices (1, 3, 5, 7, or 9).
For the tens digit, we also have 5 choices (1, 3, 5, 7, or 9).
For the units digit, we again have 5 choices (1, 3, 5, 7, or 9).
By the counting principle, the total number of three-digit positive integers with only odd digits is $5 \times 5 \times 5 = \boxed{125}$.The answer is: 125
Below is a math problem. Provide a detailed, step-by-step solution.

### Problem:
How many distinct three-digit positive integers have only odd digits?

### Answer:
