Number from 1 to 42 divisible by none of 2, 3, 7

Question

An integer is chosen uniformly at random from 1 to 42. What is the probability that it is divisible by none of 2, 3, and 7?

Accepted Answer

Count "divisible by at least one" with inclusion-exclusion, then take the complement. Among 1 to 42: - singles: multiples of 2 give 21, of 3 give 14, of 7 give 6, - pairs: multiples of 6 give 7, of 14 give 3, of 21 give 2, - triple: multiples of 42 give 1. Then \(\text{at least one}) = (21 + 14 + 6) - (7 + 3 + 2) + 1 = 41 - 12 + 1 = 30. So 42 - 30 = 12 numbers avoid all three, and the probability is \frac{12}{42} = \frac{2}{7} \approx 0.2857. Takeaway. The pair counts use the products 6, 14, 21 and the triple uses 42, since these primes are all coprime.

Number from 1 to 42 divisible by none of 2, 3, 7

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