How it works
The least common multiple of two numbers is the smallest positive number that both divide. Using prime factorizations, take every prime to its largest power: 12 = 2² × 3 and 18 = 2 × 3², so lcm = 2² × 3² = 36.
For two numbers you can also use
lcm(a, b) = a × b ÷ gcd(a, b)
Repeating-event questions ("ring every 8 and 12 minutes; when together again?") are LCM questions. For three or more numbers, take the largest power of every prime that appears in any of them.
Least Common Multiple practice problems with solutions
These problems come from Mathyessy's question bank. Try each one before opening the solution.
What is LCM of 12 and 18?
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Answer: 36
12 = 2² × 3 and 18 = 2 × 3². The largest powers are 2² and 3², so the LCM is 36.
Two bells ring every 8 minutes and every 12 minutes. If they ring together now, in how many minutes will they ring together again?
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Answer: 24 minutes
They ring together every lcm(8, 12) minutes. 8 = 2³ and 12 = 2² × 3, so lcm = 2³ × 3 = 24.
Three bells ring every 12 minutes and every 18 minutes and every 20 minutes. If they ring together now, in how many minutes will they ring together again?
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Answer: 180 minutes
12 = 2² × 3, 18 = 2 × 3², 20 = 2² × 5. The largest powers give 2² × 3² × 5 = 180.
Two numbers have GCD 6 and LCM 180. What is the smallest possible sum of the two numbers?
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Answer: 66
Write them as 6m and 6n with m, n sharing no factor and mn = 180 ÷ 6 = 30. The pairs are (1, 30), (2, 15), (3, 10), (5, 6); the smallest sum is 5 + 6 = 11, giving 30 + 36 = 66.
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Related topics
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