How many positive integers less than or equal to $1000$ are divisible by $2$ and $3$ but not by $5$?

[i]2015 CCA Math Bonanza Individual Round #6[/i]
1. **Determine the number of integers less than or equal to 1000 that are divisible by 6:**
   - Since a number divisible by both 2 and 3 is divisible by their least common multiple (LCM), which is 6, we first find the number of integers less than or equal to 1000 that are divisible by 6.
   - This can be calculated as:
     \[
     \left\lfloor \frac{1000}{6} \right\rfloor = \left\lfloor 166.6667 \right\rfloor = 166
     \]

2. **Determine the number of integers less than or equal to 1000 that are divisible by both 6 and 5:**
   - The LCM of 6 and 5 is 30. Therefore, we need to find the number of integers less than or equal to 1000 that are divisible by 30.
   - This can be calculated as:
     \[
     \left\lfloor \frac{1000}{30} \right\rfloor = \left\lfloor 33.3333 \right\rfloor = 33
     \]

3. **Subtract the number of integers divisible by 30 from the number of integers divisible by 6:**
   - We subtract the count of numbers divisible by 30 from the count of numbers divisible by 6 to exclude those that are also divisible by 5.
   - This gives us:
     \[
     166 - 33 = 133
     \]

4. **Verify the arithmetic:**
   - The arithmetic steps provided in the solution are correct, but they are overly complicated. Simplifying the subtraction directly:
     \[
     166 - 33 = 133
     \]

The final answer is \(\boxed{133}\).