Numbers up to 100 divisible by 4 or 6

Question

Consider the integers from 1 to 100. How many are divisible by 4 or by 6 (at least one of the two)?

Accepted Answer

For two sets, inclusion-exclusion says A + B - A \cap B. The overlap is the numbers divisible by both 4 and 6, which means divisible by their least common multiple, \operatorname{lcm}(4,6) = 12 (not 24, since 4 and 6 share a factor of 2). Counts using \lfloor 100/d \rfloor: - divisible by 4: 25. - divisible by 6: 16. - divisible by 12: 8. So 25 + 16 - 8 = 33. Takeaway. The overlap of "divisible by a" and "divisible by b" is divisibility by \operatorname{lcm}(a,b), not ab. Here the answer is 33.

Numbers up to 100 divisible by 4 or 6

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