To solve this problem, we need to determine how many integers between 1 and 2000 satisfy the following conditions:

1. Divisible by 8
2. Not divisible by 2
3. Not divisible by 5

Step 1: Count the integers divisible by 8

The integers divisible by 8 between 1 and 2000 can be represented as: $8, 16, 24, \dots, 2000$ This sequence can be written as: $8 \times 1, 8 \times 2, 8 \times 3, \dots, 8 \times 250$ So, there are 250 integers between 1 and 2000 that are divisible by 8.

Step 2: Analyze the additional conditions

To meet the condition of being divisible by 8 but not divisible by 2, is impossible since any number divisible by 8 is automatically divisible by 2.

Therefore, we need to reconsider the question. The actual intention seems to be finding numbers divisible by 8 but not divisible by 5 and divisible by 2 (since all numbers divisible by 8 meet this condition).

Step 3: Apply the condition of not being divisible by 5

Numbers divisible by 8 that are not divisible by 5 are simply those numbers divisible by 8 and not by $8 \times 5 = 40$.

Let's count how many multiples of 40 are in the sequence of multiples of 8 from 1 to 2000:

• The smallest multiple of 40 is $40 = 8 \times 5$.
• The largest multiple of 40 up to 2000 is $2000 = 40 \times 50$.

So, there are 50 multiples of 40 between 1 and 2000.

Step 4: Subtract the number of multiples of 40

Finally, subtract the 50 multiples of 40 from the total 250 numbers divisible by 8: $250 - 50 = 200$ So, there are 200 integers between 1 and 2000 that are divisible by 8, not divisible by 5, but divisible by 2.

Would you like further details or have any questions? Here are some related problems you might find interesting:

1. How many integers between 1 and 2000 are divisible by 4 but not by 8?
2. What is the sum of all integers between 1 and 2000 that are divisible by 8?
3. How many integers between 1 and 1000 are divisible by both 3 and 7?
4. Find the number of integers between 1 and 5000 that are divisible by 6 but not by 9.
5. How many integers between 1 and 3000 are divisible by either 7 or 11?

Tip: When dealing with divisibility problems, it's often useful to break the problem into smaller steps and consider the properties of numbers involved, like factors and multiples.