Number from 1 to 200 divisible by 5 or 8

Question

An integer is chosen uniformly at random from 1 to 200. What is the probability that it is divisible by 5 or by 8?

Accepted Answer

Count multiples of each and subtract the overlap. Among 1 to 200: - multiples of 5: 200/5 = 40, - multiples of 8: 200/8 = 25, - multiples of both. Since 5 and 8 share no common factor, a number divisible by both is a multiple of 5 \times 8 = 40: that is 200/40 = 5. Inclusion-exclusion gives \(5 \text{ or } 8) = 40 + 25 - 5 = 60, so the probability is \frac{60}{200} = \frac{3}{10}. Takeaway. When the two divisors are coprime, their least common multiple is just the product, so subtract multiples of 40.

Number from 1 to 200 divisible by 5 or 8

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