Discussion
The number of positive integers, not greater than 100 which are not divisible by 2, 3 or 5, is :
Answer: C
There are 26 such integers.
Count the numbers up to 100 divisible by 2, 3 or 5 using inclusion-exclusion:
\[ \begin{aligned} &50 + 33 + 20 \\ &- (16 + 10 + 6) + 3 = 74 \end{aligned} \]
(multiples of 2, 3, 5; then of 6, 10, 15; then of 30). So \(100 - 74 = 26\) numbers are divisible by none of them.
Check by Euler-type reasoning: in each block of 30 there are \(30(1/2)(2/3)(4/5) = 8\) such numbers, giving 24 up to 90, plus 91 and 97 between 91 and 100, a total of 26.
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