We want to find how many integers between 1 and 500 are divisible by 2, 3, and 5 but not by 7.

Step 1: Least Common Multiple (LCM)

First, let's determine the least common multiple (LCM) of 2, 3, and 5.

$\text{LCM}(2, 3, 5) = 2 \times 3 \times 5 = 30$

So, we are looking for numbers divisible by 30, but not divisible by 7.

Step 2: Count numbers divisible by 30

The total number of integers divisible by 30 between 1 and 500 can be found by dividing 500 by 30:

$\left\lfloor \frac{500}{30} \right\rfloor = 16$

So, there are 16 integers divisible by 30 between 1 and 500.

Step 3: Exclude numbers divisible by both 30 and 7

Now, let's find how many of those 16 numbers are divisible by both 30 and 7. To do this, we calculate the LCM of 30 and 7:

$\text{LCM}(30, 7) = 210$

Next, we find how many numbers between 1 and 500 are divisible by 210:

$\left\lfloor \frac{500}{210} \right\rfloor = 2$

So, there are 2 numbers divisible by both 30 and 7.

Step 4: Final count

To get the number of integers divisible by 30 but not divisible by 7, we subtract the 2 numbers divisible by 210 from the 16 numbers divisible by 30:

$16 - 2 = 14$

Thus, there are 14 positive integers between 1 and 500 that are divisible by 2, 3, and 5 but not by 7.

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

1. How many integers between 1 and 1000 are divisible by 3, 4, and 6 but not by 5?
2. Find the number of integers divisible by 2 and 3 but not by 5 between 1 and 300.
3. How many integers between 1 and 500 are divisible by both 6 and 10?
4. Find how many numbers between 1 and 1000 are divisible by 4 but not by 9.
5. How many integers between 1 and 500 are divisible by 3, 5, and 7?

Tip: When working with multiple divisors, using the least common multiple (LCM) helps efficiently count divisible numbers!