How many numbers between 1 and 2022 inclusive are not divisible be neither 3, nor 5, nor 7?

To find the number of integers between 1 and 2022 that are not divisible by 3, 5, or 7, we'll use the principle of Inclusion-Exclusion.

### Step 1: Calculate the number of integers divisible by each number

1. **Divisible by 3:**
   The number of integers divisible by 3 between 1 and 2022 is:
   \[
   \left\lfloor \frac{2022}{3} \right\rfloor = 674
   \]

2. **Divisible by 5:**
   The number of integers divisible by 5 between 1 and 2022 is:
   \[
   \left\lfloor \frac{2022}{5} \right\rfloor = 404
   \]

3. **Divisible by 7:**
   The number of integers divisible by 7 between 1 and 2022 is:
   \[
   \left\lfloor \frac{2022}{7} \right\rfloor = 288
   \]

### Step 2: Calculate the number of integers divisible by combinations of two numbers

1. **Divisible by 3 and 5 (LCM = 15):**
   \[
   \left\lfloor \frac{2022}{15} \right\rfloor = 134
   \]

2. **Divisible by 3 and 7 (LCM = 21):**
   \[
   \left\lfloor \frac{2022}{21} \right\rfloor = 96
   \]

3. **Divisible by 5 and 7 (LCM = 35):**
   \[
   \left\lfloor \frac{2022}{35} \right\rfloor = 57
   \]

### Step 3: Calculate the number of integers divisible by all three numbers

1. **Divisible by 3, 5, and 7 (LCM = 105):**
   \[
   \left\lfloor \frac{2022}{105} \right\rfloor = 19
   \]

### Step 4: Apply the principle of Inclusion-Exclusion

The total number of integers between 1 and 2022 that are divisible by 3, 5, or 7 is:
\[
N_{\text{div}} = (674 + 404 + 288) - (134 + 96 + 57) + 19 = 1366 - 287 + 19 = 1098
\]

### Step 5: Subtract from the total number of integers

The total number of integers between 1 and 2022 is:
\[
2022 - 1 + 1 = 2022
\]

Thus, the number of integers between 1 and 2022 that are not divisible by 3, 5, or 7 is:
\[
2022 - 1098 = 924
\]

So, **924** numbers between 1 and 2022 are not divisible by 3, 5, or 7.

Would you like more details or have any questions?

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**Related Questions:**
1. How many numbers between 1 and 2022 are divisible by either 3, 5, or 7?
2. How many numbers between 1 and 2022 are divisible by 2, 3, or 5?
3. What is the sum of all numbers between 1 and 2022 that are divisible by 3?
4. How many numbers between 1 and 1000 are not divisible by 4, 6, or 9?
5. How many numbers between 1 and 2022 are divisible by exactly two of the numbers 3, 5, or 7?

**Tip:** The principle of Inclusion-Exclusion is a powerful tool to calculate the size of the union of multiple sets, especially when dealing with divisibility problems.

Discover how many numbers between 1 and 2022 are not divisible by 3, 5, or 7 using the Inclusion-Exclusion Principle in number theory. Learn step-by-step how to calculate this, including examples and detailed explanations.

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