Number of Positive Integers Divisible by 5 or 7 but not Both

To find the number of positive integers up to 1000 that are divisible by either 5 or 7 but not both, we can break down the problem into three parts:
1. Count the number of positive integers divisible by 5.
2. Count the number of positive integers divisible by 7.
3. Subtract the number of positive integers divisible by both 5 and 7.

Counting Positive Integers Divisible by 5

To count the number of positive integers divisible by 5, we can divide the highest integer (1000) by 5 and round down to the nearest whole number. This gives us 200 as the number of multiples of 5 up to 1000.

Counting Positive Integers Divisible by 7

Similarly, to count the number of positive integers divisible by 7, we divide 1000 by 7 and round down to the nearest whole number, giving us 142.

Counting Positive Integers Divisible by Both 5 and 7

To find the number of positive integers divisible by both 5 and 7, we need to find the common multiples of 5 and 7. The least common multiple (LCM) of 5 and 7 is 35. To find how many multiples of 35 are there up to 1000, we divide 1000 by 35 and round down to the nearest whole number, which is 28.

Subtracting the Number of Integers Divisible by Both 5 and 7

To get the final count of positive integers that are divisible by either 5 or 7 but not both, we subtract the count of integers divisible by both 5 and 7 from the sum of counts of integers divisible by 5 and 7.

200 (divisible by 5) + 142 (divisible by 7) - 28 (divisible by both 5 and 7) = 314 - 28 = 286

Therefore, the correct answer is 286, which represents the number of positive integers up to 1000 that are divisible by either 5 or 7 but not both.

314