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.

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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.