Number from 1 to 100 divisible by 2 or 5

Question

An integer is chosen uniformly at random from 1 to 100. What is the probability that it is divisible by 2 or by 5?

Accepted Answer

Count the multiples of each, then remove the double count. Among 1 to 100: - multiples of 2: 100/2 = 50, - multiples of 5: 100/5 = 20, - multiples of both, meaning multiples of 10: 100/10 = 10. A number divisible by both 2 and 5 was counted in the first list and again in the second, so inclusion-exclusion subtracts it once: \(2 \text{ or } 5) = 50 + 20 - 10 = 60. The probability is \frac{60}{100} = \frac{3}{5}. Takeaway. To count multiples of a or b, add the counts and subtract the multiples of their least common multiple, which here is 10.

Number from 1 to 100 divisible by 2 or 5

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