How many positive integers less that $200$ are relatively prime to either $15$ or $24$?
To find the number of positive integers less than $200$ that are relatively prime to either $15$ or $24$, we need to count the integers that are not divisible by $2$, $3$, or $5$. We will use the principle of inclusion-exclusion (PIE) to solve this problem.

1. **Count the multiples of $2$, $3$, and $5$ less than $200$:**
   - The number of integers less than $200$ and divisible by $2$ is:
     \[
     \left\lfloor \frac{200}{2} \right\rfloor = 100
     \]
   - The number of integers less than $200$ and divisible by $3$ is:
     \[
     \left\lfloor \frac{200}{3} \right\rfloor = 66
     \]
   - The number of integers less than $200$ and divisible by $5$ is:
     \[
     \left\lfloor \frac{200}{5} \right\rfloor = 40
     \]

2. **Count the multiples of the least common multiples (LCM) of pairs of $2$, $3$, and $5$:**
   - The number of integers less than $200$ and divisible by $6$ (LCM of $2$ and $3$) is:
     \[
     \left\lfloor \frac{200}{6} \right\rfloor = 33
     \]
   - The number of integers less than $200$ and divisible by $10$ (LCM of $2$ and $5$) is:
     \[
     \left\lfloor \frac{200}{10} \right\rfloor = 20
     \]
   - The number of integers less than $200$ and divisible by $15$ (LCM of $3$ and $5$) is:
     \[
     \left\lfloor \frac{200}{15} \right\rfloor = 13
     \]

3. **Count the multiples of the LCM of $2$, $3$, and $5$:**
   - The number of integers less than $200$ and divisible by $30$ (LCM of $2$, $3$, and $5$) is:
     \[
     \left\lfloor \frac{200}{30} \right\rfloor = 6
     \]

4. **Apply the principle of inclusion-exclusion (PIE):**
   - The number of integers less than $200$ that are divisible by $2$, $3$, or $5$ is:
     \[
     100 + 66 + 40 - 33 - 20 - 13 + 6 = 146
     \]

5. **Subtract from the total number of integers less than $200$:**
   - The total number of integers less than $200$ is $199$ (since we are considering positive integers).
   - Therefore, the number of integers less than $200$ that are relatively prime to $2$, $3$, and $5$ is:
     \[
     199 - 146 = 53
     \]

The final answer is $\boxed{53}$.