Number from 1 to 60 divisible by 4 or 6

Question

An integer is chosen uniformly at random from 1 to 60. What is the probability that it is divisible by 4 or by 6?

Accepted Answer

Here the two divisors share a factor, so be careful with the overlap. Among 1 to 60: - multiples of 4: 60/4 = 15, - multiples of 6: 60/6 = 10, - multiples of both. A number divisible by 4 and by 6 is a multiple of \operatorname{lcm}(4,6) = 12, not 24. So the count is 60/12 = 5. Inclusion-exclusion gives \(4 \text{ or } 6) = 15 + 10 - 5 = 20, and the probability is \frac{20}{60} = \frac{1}{3}. Takeaway. When divisors overlap, subtract multiples of their least common multiple, not the product. Here \operatorname{lcm}(4,6)=12, while 4 \times 6 = 24 would be wrong.

Number from 1 to 60 divisible by 4 or 6

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