Problem: Determine the number of positive integers which are not divisible by any of 2, 3, or 5.

Solution:
To find the number of positive integers that are not divisible by 2, 3, or 5, we can use the principle of inclusion-exclusion. We will subtract the numbers that are divisible by 2, 3, or 5 from the total count of positive integers.

Step 1: Find the total count of positive integers:
We need to determine the total number of positive integers. There is no upper limit specified, so we will assume it to be infinity. Therefore, the total count of positive integers is infinite.

Step 2: Find the count of positive integers divisible by 2:
To find the count of positive integers divisible by 2, we divide the total count of positive integers by 2. However, since we assumed the total count to be infinite, the count of positive integers divisible by 2 is also infinite.

Step 3: Find the count of positive integers divisible by 3:
To find the count of positive integers divisible by 3, we divide the total count of positive integers by 3. Again, the count is infinite.

Step 4: Find the count of positive integers divisible by 5:
To find the count of positive integers divisible by 5, we divide the total count of positive integers by 5. The count is infinite.

Step 5: Apply the principle of inclusion-exclusion:
To find the count of positive integers that are not divisible by any of 2, 3, or 5, we subtract the counts from steps 2, 3, and 4 from the total count of positive integers.

Total count - (Count divisible by 2 + Count divisible by 3 + Count divisible by 5)

Infinite - (Infinite + Infinite + Infinite) = Infinite

Therefore, there are an infinite number of positive integers that are not divisible by any of 2, 3, or 5.

However, it seems there might be some misunderstanding or error in the given correct answer '720'. It is not possible to have a finite count for positive integers not divisible by any of 2, 3, or 5.