Determine the number of positive integer n, 1 ≤ 𝑛 ≤ 2000 that are not divisible
by 2, 3 or 5 but are divisible by 7

We need to determine the number of positive integers \( n \) such that \( 1 \leq n \leq 2000 \) that are not divisible by 2, 3, or 5, but are divisible by 7.

### Step 1: Identify numbers divisible by 7
First, we find the range of numbers divisible by 7 within the given range \( 1 \leq n \leq 2000 \).

The smallest number divisible by 7 is \( 7 \times 1 = 7 \).
The largest number divisible by 7 and less than or equal to 2000 is \( 7 \times 285 = 1995 \).

So, the numbers divisible by 7 within the range are:
\[ 7, 14, 21, \ldots, 1995 \]

The number of terms in this sequence is:
\[
n = \frac{1995 - 7}{7} + 1 = 285
\]

So, there are 285 numbers divisible by 7 in the range.

### Step 2: Remove numbers divisible by 2, 3, or 5
Next, we need to subtract the numbers divisible by 2, 3, or 5 from this set.

#### Numbers divisible by both 7 and 2:
These numbers are divisible by \( 7 \times 2 = 14 \).
The smallest is \( 14 \times 1 = 14 \) and the largest is \( 14 \times 142 = 1988 \).
The number of such terms is:
\[
\frac{1988 - 14}{14} + 1 = 142
\]

#### Numbers divisible by both 7 and 3:
These numbers are divisible by \( 7 \times 3 = 21 \).
The smallest is \( 21 \times 1 = 21 \) and the largest is \( 21 \times 95 = 1995 \).
The number of such terms is:
\[
\frac{1995 - 21}{21} + 1 = 95
\]

#### Numbers divisible by both 7 and 5:
These numbers are divisible by \( 7 \times 5 = 35 \).
The smallest is \( 35 \times 1 = 35 \) and the largest is \( 35 \times 57 = 1995 \).
The number of such terms is:
\[
\frac{1995 - 35}{35} + 1 = 57
\]

### Step 3: Apply the Inclusion-Exclusion Principle
Now, we need to account for overlaps between these sets.

#### Numbers divisible by 7, 2, and 3:
These numbers are divisible by \( 7 \times 2 \times 3 = 42 \).
The smallest is \( 42 \times 1 = 42 \) and the largest is \( 42 \times 47 = 1974 \).
The number of such terms is:
\[
\frac{1974 - 42}{42} + 1 = 47
\]

#### Numbers divisible by 7, 2, and 5:
These numbers are divisible by \( 7 \times 2 \times 5 = 70 \).
The smallest is \( 70 \times 1 = 70 \) and the largest is \( 70 \times 28 = 1960 \).
The number of such terms is:
\[
\frac{1960 - 70}{70} + 1 = 28
\]

#### Numbers divisible by 7, 3, and 5:
These numbers are divisible by \( 7 \times 3 \times 5 = 105 \).
The smallest is \( 105 \times 1 = 105 \) and the largest is \( 105 \times 19 = 1995 \).
The number of such terms is:
\[
\frac{1995 - 105}{105} + 1 = 19
\]

#### Numbers divisible by 7, 2, 3, and 5:
These numbers are divisible by \( 7 \times 2 \times 3 \times 5 = 210 \).
The smallest is \( 210 \times 1 = 210 \) and the largest is \( 210 \times 9 = 1890 \).
The number of such terms is:
\[
\frac{1890 - 210}{210} + 1 = 9
\]

### Final Calculation:
Using the Inclusion-Exclusion Principle:
\[
\text{Total Count} = 285 - (142 + 95 + 57) + (47 + 28 + 19) - 9
\]
\[
\text{Total Count} = 285 - 294 + 94 - 9 = 76
\]

Thus, the number of positive integers \( n \) such that \( 1 \leq n \leq 2000 \), not divisible by 2, 3, or 5, but divisible by 7 is **76**.

Would you like to see more details or have any questions?

### Related Questions:
1. What is the number of integers divisible by 7 and 11 but not by 3 within the range 1 to 1000?
2. How many numbers less than 2000 are divisible by 2 and 3 but not by 7?
3. Can you calculate the number of integers divisible by both 6 and 9 within the range of 1 to 1500?
4. What is the number of positive integers less than 2000 that are divisible by 11 and not divisible by 2 or 5?
5. How many numbers between 1 and 3000 are divisible by 5 but not by 2 or 3?

### Tip:
When dealing with problems of divisibility, using the Inclusion-Exclusion Principle helps accurately count elements that satisfy multiple conditions.

Explore how to count positive integers n such that 1 ≤ n ≤ 2000, not divisible by 2, 3, or 5, but divisible by 7. Learn the application of the Inclusion-Exclusion Principle to solve this number theory problem effectively.

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