Count the multiples of 3, 17, and 35 in the given range.
We have
1000 = 999 + 1 = 3•333 + 1
so there are 333 multiples of 3;
1000 = 986 + 14 = 17•58 + 14
so there are 58 multiples of 17; and
1000 = 980 + 20 = 35•28 + 20
so there are 28 multiples of 35.
Now count the multiples of 3 and 17, 3 and 35, and 17 and 35.
Since LCM(3, 17) = 51, we have
1000 = 969 + 31 = 51•19 + 31
so there are 19 multiples of 51.
LCM(3, 35) = 105, and
1000 = 945 + 55 = 105•9 + 55
so there are 9 multiples of 105.
LCM(17, 35) = 595, and
1000 = 595 + 405
so there is only 1 multiple of 595.
Also count the multiples of 3 and 17 and 35 together.
LCM(3, 17, 35) = 1785 which exceed 1000, so there are no more multiples.
Now use the inclusion/exclusion principle to count the multiples of 3 or 17 or 35 or any combination of them.
333 + 58 + 28 - (19 + 9 + 1) = 390
Then there are 1000 - 390 = 610 positive integers in the range that are not divisible by 3, 17, or 35.