How many integers between 1 and 2022 inclusive are not divisible by neither 3 nor 5 nor 7?

To solve this problem, we need to find how many integers between 1 and 2022 are not divisible by 3, 5, or 7. We can use the principle of inclusion and exclusion (PIE) to do this efficiently.

### Step 1: Find the Total Number of Integers
The total number of integers between 1 and 2022 is simply 2022.

### Step 2: Find the Number of Multiples of 3, 5, and 7
- Multiples of 3: \( \left\lfloor \frac{2022}{3} \right\rfloor = 674 \)
- Multiples of 5: \( \left\lfloor \frac{2022}{5} \right\rfloor = 404 \)
- Multiples of 7: \( \left\lfloor \frac{2022}{7} \right\rfloor = 288 \)

### Step 3: Find the Number of Multiples of Pairs of Numbers (Using the Principle of Inclusion and Exclusion)
- Multiples of \(3 \times 5 = 15\): \( \left\lfloor \frac{2022}{15} \right\rfloor = 134 \)
- Multiples of \(3 \times 7 = 21\): \( \left\lfloor \frac{2022}{21} \right\rfloor = 96 \)
- Multiples of \(5 \times 7 = 35\): \( \left\lfloor \frac{2022}{35} \right\rfloor = 57 \)

### Step 4: Find the Number of Multiples of All Three Numbers
- Multiples of \(3 \times 5 \times 7 = 105\): \( \left\lfloor \frac{2022}{105} \right\rfloor = 19 \)

### Step 5: Apply the Principle of Inclusion and Exclusion
Using the principle of inclusion and exclusion, the number of integers divisible by 3, 5, or 7 is:

\[
N(3 \cup 5 \cup 7) = (674 + 404 + 288) - (134 + 96 + 57) + 19 = 1366
\]

### Step 6: Subtract from the Total
The number of integers between 1 and 2022 that are **not** divisible by 3, 5, or 7 is:

\[
2022 - 1366 = 656
\]

Thus, there are **656 integers** between 1 and 2022 that are not divisible by 3, 5, or 7.

Would you like more details on any of these steps or have any questions?

Here are 5 related questions:
1. How can we apply the inclusion-exclusion principle to three or more sets?
2. What is the probability that a randomly chosen integer between 1 and 2022 is divisible by either 3, 5, or 7?
3. How many integers between 1 and 10000 are divisible by either 3, 5, or 7?
4. How would the result change if we also included divisibility by 11?
5. Can we use Venn diagrams to visually represent the inclusion-exclusion principle for this problem?

**Tip:** When using the inclusion-exclusion principle, always be careful to subtract overlapping pairs of sets and then add back in the overlapping triplets to avoid overcounting.

Discover how to count integers between 1 and 2022 that are not divisible by 3, 5, or 7 using the Principle of Inclusion and Exclusion. Learn step-by-step with detailed calculations and examples. Suitable for advanced high school students.

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