Final answer:
There are 461 positive integers less than or equal to 2015 that are divisible by 3 but not by 5 or 7. We find this by initially counting all multiples of 3 and then subtracting those also divisible by 5, 7, and both (multiples of 15, 21, and 105 respectively).
Step-by-step explanation:
The question asks how many positive integers less than or equal to 2015 are divisible by 3 but are neither divisible by 5 nor 7. To find this number, we can first determine how many numbers less than or equal to 2015 are divisible by 3. We do this by dividing 2015 by 3, which gives us 671.33, so there are 671 multiples of 3 up to 2015 (since we consider only whole numbers).
Next, we need to exclude the numbers that are also divisible by 5 or 7. We find the number of multiples of 15 (3*5) up to 2015 by dividing 2015 by 15, which is 134, and the number of multiples of 21 (3*7) by dividing 2015 by 21, giving us 95.9, so 95 numbers.
However, some numbers are multiples of both 15 and 21 (multiples of 105, since 15*7 = 105), and we should only count them once. There are 19 multiples of 105 up to 2015 (2015/105). Therefore, the total number we're looking for is all multiples of 3 minus multiples of 15 and 21, plus those multiples of 105 we removed twice: 671 - 134 - 95 + 19 = 461.
So, there are 461 positive integers less than or equal to 2015 that are divisible by 3 but not by 5 or 7.