1. **Stating the problem:** We are given two events A and B with probabilities P(A) and P(B), and the formula for the probability of both events occurring together: $$P(A \text{ and } B) = P(A) \cdot P(B).$$ We need to find the value of $m$ given that $m$ and $n$ are related by the problem and the answer is a positive integer. 2. **Understanding the formula:** The formula $$P(A \text{ and } B) = P(A) \cdot P(B)$$ applies when events A and B are independent. This means the occurrence of one does not affect the probability of the other. 3. **Given:** The problem states $m$ and $n$ with a fraction $$\frac{m}{n}$$ and the relation involving $m$ and $n$ is implied by the problem context. 4. **Assuming:** Since the problem is labeled "Problem 25" and the answer is a positive integer, and given the formula, we infer that $m$ and $n$ relate to probabilities such that $$P(A) = \frac{m}{n}$$ and $$P(B) = \frac{m}{n}$$ or similar. 5. **Solving:** Using the independence formula, $$P(A \text{ and } B) = P(A) \cdot P(B) = \frac{m}{n} \cdot \frac{m}{n} = \frac{m^2}{n^2}.$$ 6. **Since the answer is a positive integer,** we set $$\frac{m^2}{n^2} = k,$$ where $k$ is a positive integer. 7. **Therefore,** $$m^2 = k n^2,$$ which implies $$m = n \sqrt{k}.$$ For $m$ to be an integer, $\sqrt{k}$ must be rational, and since $k$ is a positive integer, $k$ must be a perfect square. 8. **Conclusion:** The value of $m$ is a positive integer multiple of $n$ times the square root of a perfect square integer $k$. Without additional information, the simplest positive integer solution is when $k=1$, so $$m = n.$$ **Final answer:** $$m = n.$$