How can the number of numbers from 0 to 999 which are not divisible by either 5 or 7 be found?

We need to find the number of values in the set of numbers
from 0 to 999 which are not divisible by 5 or 7.

The
largest multiple of 5 just smaller than 999 is 995 = 199*5 and the largest multiple of 7
just smaller than 999 is 994 = 142*7

So we have 199 numbers
divisible by 5 and 142 numbers divisible by 7. But here we have counted the numbers
divisible by both 5 and 7 twice. The largest multiple of 35 less than 999 is 980 =
35*28

So there are 142 + 199 - 28 = 313 numbers divisible
by 5 or 7 in the given set of numbers from 0 to 999.

The
other numbers are not divisible by either 5 or 7. There are 1000 - 313 = 687 such
numbers.

The required number of numbers not
divisible by 5 or 7 in the set of numbers from 0 to 999 is
687.