Given $N = 2001$, let's calculate how many positive integers not exceeding 2001 are multiples of 3 or 4 but not 5.

Step 1: Calculate the relevant multiples

1. Multiples of 3 up to 2001: $\left\lfloor \frac{2001}{3} \right\rfloor = \left\lfloor 667 \right\rfloor = 667$

2. Multiples of 4 up to 2001: $\left\lfloor \frac{2001}{4} \right\rfloor = \left\lfloor 500.25 \right\rfloor = 500$

3. Multiples of 12 up to 2001 (common multiples of 3 and 4): $\left\lfloor \frac{2001}{12} \right\rfloor = \left\lfloor 166.75 \right\rfloor = 166$

Using the inclusion-exclusion principle: $\text{Multiples of 3 or 4} = 667 + 500 - 166 = 1001$

Step 2: Subtract the multiples of 5

4. Multiples of 5 up to 2001: $\left\lfloor \frac{2001}{5} \right\rfloor = \left\lfloor 400.2 \right\rfloor = 400$

5. Multiples of 15 up to 2001 (multiples of both 3 and 5): $\left\lfloor \frac{2001}{15} \right\rfloor = \left\lfloor 133.4 \right\rfloor = 133$

6. Multiples of 20 up to 2001 (multiples of both 4 and 5): $\left\lfloor \frac{2001}{20} \right\rfloor = \left\lfloor 100.05 \right\rfloor = 100$

7. Multiples of 60 up to 2001 (multiples of 3, 4, and 5): $\left\lfloor \frac{2001}{60} \right\rfloor = \left\lfloor 33.35 \right\rfloor = 33$

Now, subtracting the multiples of 5: $\text{Multiples of 3 or 4 but not 5} = 1001 - (133 + 100 - 33) = 1001 - 200 = 801$

Final Answer:

There are 801 positive integers not exceeding 2001 that are multiples of 3 or 4 but not 5.

Would you like more details or have any questions?

---

Here are 5 related questions:

1. How would the count change if we considered numbers up to 5000?
2. What would be the count if we included multiples of 6 as well?
3. How does the inclusion-exclusion principle help avoid over-counting?
4. Can you calculate the number of multiples of 3 and 4 but not multiples of 2?
5. What is the significance of finding common multiples, like those of 12, 15, and 20, in this problem?

Tip: Double-check your work by re-evaluating edge cases, such as considering the largest multiple within the given range. This helps ensure accuracy, especially when using the inclusion-exclusion principle.