Number from 1 to 120 divisible by 2 or 3

Question

An integer is chosen uniformly at random from 1 to 120. What is the probability that it is divisible by 2 or by 3?

Accepted Answer

Count multiples of each and subtract the overlap. Among 1 to 120: - multiples of 2: 120/2 = 60, - multiples of 3: 120/3 = 40, - multiples of both, meaning multiples of 6: 120/6 = 20. Inclusion-exclusion gives \(2 \text{ or } 3) = 60 + 40 - 20 = 80, so the probability is \frac{80}{120} = \frac{2}{3}. Takeaway. Add the two multiple counts and subtract the multiples of \operatorname{lcm}(2,3) = 6.

Number from 1 to 120 divisible by 2 or 3

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