Numbers up to 300 divisible by 6, 10, or 15

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Consider the integers from 1 to 300. How many are divisible by 6, by 10, or by 15 (at least one of these)?

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Use inclusion-exclusion on three sets, remembering overlaps use least common multiples. Single counts (using \lfloor 300/d \rfloor): - by 6: 50; by 10: 30; by 15: 20. Pair overlaps: here every pair's lcm equals 30: - \operatorname{lcm}(6,10) = 30, \operatorname{lcm}(6,15) = 30, \operatorname{lcm}(10,15) = 30, each giving \lfloor 300/30 \rfloor = 10. Triple overlap: \operatorname{lcm}(6,10,15) = 30, also 10. Combine: 50 + 30 + 20 - (10 + 10 + 10) + 10 = 100 - 30 + 10 = 80. Takeaway. Overlaps depend on least common multiples, which can coincide: here all three pairwise lcms and the triple lcm a…

Numbers up to 300 divisible by 6, 10, or 15

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