How many positive integers not exceeding  are multiples of 3 or 4 but not 5?

Let's solve the problem step by step.

### Step 1: Define the problem
We need to find how many positive integers \( n \) not exceeding a certain value (the upper bound wasn't provided, so let's assume it is \( N \)) are multiples of 3 or 4 but not 5.

### Step 2: Calculate multiples of 3, 4, and 5
- **Multiples of 3**: The number of multiples of 3 up to \( N \) is given by \( \left\lfloor \frac{N}{3} \right\rfloor \).
- **Multiples of 4**: The number of multiples of 4 up to \( N \) is given by \( \left\lfloor \frac{N}{4} \right\rfloor \).
- **Multiples of 5**: The number of multiples of 5 up to \( N \) is given by \( \left\lfloor \frac{N}{5} \right\rfloor \).

### Step 3: Use the inclusion-exclusion principle
To count multiples of 3 or 4, we use:
\[
\text{Multiples of 3 or 4} = \left\lfloor \frac{N}{3} \right\rfloor + \left\lfloor \frac{N}{4} \right\rfloor - \left\lfloor \frac{N}{12} \right\rfloor
\]
where \( \left\lfloor \frac{N}{12} \right\rfloor \) subtracts the overlap, i.e., the multiples of both 3 and 4.

### Step 4: Subtract multiples of 5
Next, we subtract the multiples of 5 from this count. However, we must also consider the numbers that are multiples of 3 or 4 **and** 5 (i.e., the multiples of 15 and 20, respectively):
\[
\text{Multiples of 3 or 4 but not 5} = \left(\left\lfloor \frac{N}{3} \right\rfloor + \left\lfloor \frac{N}{4} \right\rfloor - \left\lfloor \frac{N}{12} \right\rfloor\right) - \left(\left\lfloor \frac{N}{15} \right\rfloor + \left\lfloor \frac{N}{20} \right\rfloor - \left\lfloor \frac{N}{60} \right\rfloor\right)
\]
Here, \( \left\lfloor \frac{N}{60} \right\rfloor \) accounts for the overlap between 15 and 20.

### Final formula:
Thus, the formula becomes:
\[
\text{Count} = \left\lfloor \frac{N}{3} \right\rfloor + \left\lfloor \frac{N}{4} \right\rfloor - \left\lfloor \frac{N}{12} \right\rfloor - \left(\left\lfloor \frac{N}{15} \right\rfloor + \left\lfloor \frac{N}{20} \right\rfloor - \left\lfloor \frac{N}{60} \right\rfloor\right)
\]

If you provide a specific value for \( N \), I can calculate the exact number. Would you like to proceed with a specific number, or do you have any other questions?

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Here are 5 questions related to this topic:

1. What is the Inclusion-Exclusion Principle and how does it apply to this problem?
2. How would the solution change if we included multiples of 5?
3. Can this approach be extended to multiples of other numbers, such as 7 or 8?
4. What if we needed to find numbers that are multiples of 3, 4, **and** 5?
5. How would the approach change if we only wanted to count even multiples?

**Tip:** When dealing with multiples and counting, always consider overlaps between sets to avoid double-counting, as demonstrated by the Inclusion-Exclusion Principle.

Learn how to calculate the number of positive integers not exceeding N that are multiples of 3 or 4 but not 5 using the Inclusion-Exclusion Principle. This problem involves number theory concepts and practical applications of counting multiples.

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