Problem (Number Theory):
Determine the number of positive integers less than or equal to 300 that are multiples of 3 or 5 , but are not multiples of 10 or 15 .

Solution:
Since the least common multiple of $3,5,10$ and 15 is 30 , then we can count the number of positive integers less than or equal to 30 satisfying these conditions, and multiply the total by 10 to obtain the number less than 300. (This is because each group of 30 consecutive integers starting with 1 more than a multiple of 30 will have the same number of integers having these properties, because we can subtract 30 from each one and not change these properties.)

So from 1 to 30, we have:

$$
3,5,6,9,12,18,21,24,25,27
$$

Thus there are 10 less than or equal to 30 , and so 100 such positive integers less than or equal to 300 .

We proceed by doing a (careful!) count.

The number of positive multiples of 3 less than or equal to 300 is 100.

The number of positive multiples of 5 less than or equal to 300 is 60 .

Thus, we have 160 candidates, but have included multiples of 15 twice (since 15 is a multiple of each of 3 and 5), and have also included multiples of 10.

The number of multiples of 15 less than or equal to 300 is 20 , so to remove the multiples of 15 , we must remove 40 from 160 to get 120 positive integers less than or equal to 300 which are multiples of 3 or 5 but not of 15 .



This total still included some multiples of 10 that are less or equal to 300 (but not all, since we have already removed 30 , for instance).

In fact, there are 30 multiples of 10 less than or equal 300,10 of which are multiples of 15 as well (that is, the multiples of 30). So we must remove 20 from the total of 120. We then obtain that there are 100 positive integers less than or equal to 300 which are multiples of 3 or 5 , but not of 10 or 15 .

Final answer: 100