Determine the no of positive integers less than 2000 which are not divisible by 2,3,5 but divisible by 7

To determine how many positive integers less than 2000 are divisible by 7 but not divisible by 2, 3, or 5, we can break the problem down into steps:

### Step 1: Count the number of integers divisible by 7
The number of integers less than 2000 divisible by 7 is found by:
\[
\left\lfloor \frac{1999}{7} \right\rfloor = 285
\]
Thus, there are 285 integers divisible by 7.

### Step 2: Exclude integers divisible by 2, 3, or 5
Next, we need to remove the integers divisible by 7 and any of 2, 3, or 5. To do this, we use the principle of inclusion-exclusion.

#### Step 2.1: Count multiples of both 7 and 2, 3, or 5
We will count integers divisible by:
- \(7 \times 2 = 14\),
- \(7 \times 3 = 21\),
- \(7 \times 5 = 35\).

- **Multiples of 14** less than 2000:
\[
\left\lfloor \frac{1999}{14} \right\rfloor = 142
\]

- **Multiples of 21** less than 2000:
\[
\left\lfloor \frac{1999}{21} \right\rfloor = 95
\]

- **Multiples of 35** less than 2000:
\[
\left\lfloor \frac{1999}{35} \right\rfloor = 57
\]

#### Step 2.2: Remove over-counted multiples of combinations of 2, 3, and 5
Now we need to remove integers divisible by combinations of \(7 \times 2 \times 3 = 42\), \(7 \times 2 \times 5 = 70\), and \(7 \times 3 \times 5 = 105\).

- **Multiples of 42**:
\[
\left\lfloor \frac{1999}{42} \right\rfloor = 47
\]

- **Multiples of 70**:
\[
\left\lfloor \frac{1999}{70} \right\rfloor = 28
\]

- **Multiples of 105**:
\[
\left\lfloor \frac{1999}{105} \right\rfloor = 19
\]

#### Step 2.3: Add back over-counted multiples of all 2, 3, and 5
We also need to add back the number divisible by \(7 \times 2 \times 3 \times 5 = 210\), because these have been subtracted too many times.

- **Multiples of 210**:
\[
\left\lfloor \frac{1999}{210} \right\rfloor = 9
\]

### Step 3: Apply Inclusion-Exclusion
Now, apply the principle of inclusion-exclusion to count how many multiples of 7 are divisible by 2, 3, or 5:

\[
142 + 95 + 57 - (47 + 28 + 19) + 9 = 294 - 94 + 9 = 209
\]

### Step 4: Subtract from total multiples of 7
The number of integers divisible by 7 but **not divisible** by 2, 3, or 5 is:
\[
285 - 209 = 76
\]

Thus, the number of positive integers less than 2000 that are divisible by 7 but not divisible by 2, 3, or 5 is **76**.

Would you like more details or have any questions? Here are 5 related questions for practice:

1. How many integers less than 1000 are divisible by 7 but not divisible by 2, 3, or 5?
2. What is the sum of all integers less than 2000 that are divisible by 7 but not divisible by 2, 3, or 5?
3. Find how many integers less than 5000 are divisible by 7 but not divisible by 2, 3, or 5.
4. How many integers less than 2000 are divisible by 2, 3, or 5?
5. How many integers less than 2000 are divisible by 7, 11, or 13?

**Tip:** The principle of inclusion-exclusion helps solve problems where multiple conditions must be met or excluded simultaneously.

Determine the number of positive integers less than 2000 which are not divisible by 2, 3, 5 but divisible by 7.

Find how many integers less than 2000 are divisible by 7 but not divisible by 2, 3, or 5 using the Inclusion-Exclusion Principle. This step-by-step solution covers core number theory concepts suitable for Grades 9-12.

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