The correct answer is **B. 2**. To determine the number of integers $n$ such that $100 < n < 200$ and $n$ has exactly three positive factors, we follow these steps: 1. **Understand the property of integers with exactly three factors:** An integer has exactly three positive factors if and only if it is the square of a prime number. Let's prove this: * If a number $N$ has the prime factorization $N = p_1^{a_1} p_2^{a_2} \dots p_k^{a_k}$, then the number of positive factors of $N$ is given by $(a_1+1)(a_2+1)\dots(a_k+1)$. * We are given that the number of factors is 3. Since 3 is a prime number, the only way for the product $(a_1+1)(a_2+1)\dots(a_k+1)$ to be 3 is if there is only one prime factor ($k=1$) and its exponent plus one is 3. * So, $a_1+1 = 3 \Rightarrow a_1 = 2$. This means the number $N$ must be of the form $p^2$, where $p$ is a prime number. 2. **Apply the given range condition:** We need to find prime numbers $p$ such that $p^2$ falls within the range $100 < n < 200$. This translates to: $100 < p^2 < 200$ 3. **Determine the range for $p$:** * Taking the square root of the inequalities, we get: $\sqrt{100} < p < \sqrt{200}$ * Calculating the square roots: $10 < p < \sqrt{200}$ * To estimate $\sqrt{200}$, we know that $14^2 = 196$ and $15^2 = 225$. Therefore, $14 < \sqrt{200} < 15$, specifically $\sqrt{200} \approx 14.14$. * So, the range for $p$ is $10 < p < 14.14$. 4. **Identify prime numbers within this range:** We need to list all prime numbers $p$ that satisfy $10 < p < 14.14$. The prime numbers in this interval are: * $p = 11$ (since 11 is prime and $10 < 11 < 14.14$) * $p = 13$ (since 13 is prime and $10 < 13 < 14.14$) 5. **Calculate the corresponding values of $n$:** * For $p=11$, $n = 11^2 = 121$. * For $p=13$, $n = 13^2 = 169$. 6. **Verify if these values of $n$ satisfy the original condition:** * For $n=121$: $100 < 121 < 200$. This is true. * For $n=169$: $100 < 169 < 200$. This is true. 7. **Count the number of such integers:** We found two such integers: 121 and 169. Thus, there are exactly 2 integers $n$ such that $100 < n < 200$ and $n$ has exactly three positive factors.