Final answer:
The number of positive integers <= 1000 undivisible by 3, 8, or 25 is calculated by subtracting those divisible by these numbers and adding back those counted more than once due to multiple divisibility, using the principle of inclusion-exclusion.
Step-by-step explanation:
The student's question is about finding the number of positive integers less than or equal to 1000 that are not divisible by 3, 8, or 25. To calculate this, one must find the count of all numbers in the range and subtract those that are divisible by 3, 8, or 25.
However, during this process, we need to be careful to add back the numbers that were subtracted more than once because they are divisible by a combination of these divisors (like 3 and 8, or 8 and 25, etc.).
Finding the least common multiples (LCM) for these combinations is essential to adjust the count correctly. Once you have the counts for each of these categories, you can perform the necessary inclusion-exclusion principle to find the answer.