Consider the first 100 natural numbers. How many of them are not divisible by any one of 2, 3, 5, 7 and 9 ?

We need numbers from 1 to 100 not divisible by 2, 3, 5, or 7 (note: any number divisible by 9 is also divisible by 3, so 9 adds nothing new). These are numbers coprime to 2×3×5×7 = 210. Using inclusion-exclusion on {2,3,5,7}: count of numbers ≤100 divisible by at least one = 50+33+20+14 − (16+10+7+6+4+2) + (3+2+1+0) − 0 = 117 − 45 + 6 = 78. So count NOT divisible by any = 100 − 78 = 22. The number 1 is included as it is divisible by none. Hence the answer is 22.