How many numbers from 0 to 999 are divisible by either 5 or 7?

To find how many numbers from 0 to 999 are divisible by 5 or 7, we apply the Inclusion-Exclusion Principle for the range 1–999:

• Multiples of 5: 999 ÷ 5 = 199.8 ≈ 199 numbers.
• Multiples of 7: 999 ÷ 7 = 142.7 ≈ 142 numbers.
• Multiples of both (35): 999 ÷ 35 = 28.5 ≈ 28 numbers.

The total count of numbers divisible by either 5 or 7 is calculated as:

Count = n(5) + n(7) - n(5 × 7)
Count = 199 + 142 - 28 = 313.

If 0 was included as a multiple, the count would be 314, but since 313 is the available option, the question refers to the set of positive integers up to 999. Option C (686) is the number of integers not divisible by 5 or 7 (999 - 313 = 686).