Divisible by none of two, three, or five

Question

You pick a whole number uniformly at random from 1 to 30. What is the probability that it is divisible by none of 2, 3, or 5?

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Count how many numbers from 1 to 30 are multiples of at least one of 2, 3, 5, using inclusion-exclusion, then subtract from 30. Multiples of each, within 1 to 30: \lfloor 30/2 \rfloor = 15,\quad \lfloor 30/3 \rfloor = 10,\quad \lfloor 30/5 \rfloor = 6. Multiples of each pair's product (to fix the double counting): \lfloor 30/6 \rfloor = 5,\quad \lfloor 30/10 \rfloor = 3,\quad \lfloor 30/15 \rfloor = 2. Multiples of 2 \cdot 3 \cdot 5 = 30: \lfloor 30/30 \rfloor = 1. By inclusion-exclusion, the count divisible by at least one is (15 + 10 + 6) - (5 + 3 + 2) + 1 = 31 - 10 + 1 = 22. So 30 - 22 = 8…

Divisible by none of two, three, or five

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