Least Common Multiple (LCM)
In short: The least common multiple of two numbers is the first positive count that lands on both of their skip-counting lists, which is 42 for 14 and 21.
6th GradeNumbers and Operations6.NS.B.4
What is Least Common Multiple?
The least common multiple, short form LCM, is where two skip-counting lists first collide: count by 14s and by 21s side by side and both reach 42. It says when two repeating events sync up again, and which denominator two fractions can share.
Formally, a positive whole number \(m\) is the LCM of \(a\) and \(b\) if it is a multiple of \(a\) and a multiple of \(b\), and no smaller positive number is a multiple of both.
How to spot it
- Two schedules repeat at different intervals and you are asked when they happen together next.
- Fractions with unlike denominators must be rewritten over a common denominator.
- A problem looks for the smallest quantity that splits evenly into two different group sizes.
- The words “least”, “smallest” or “first time” appear beside “common multiple”.
The method
- List the first several multiples of each number: 14, 28, 42, 56 and 21, 42, 63.
- Look for the first value that appears in both lists.
- Or write the prime factorizations and take each prime to its highest power: \(14 = 2 \cdot 7\) and \(21 = 3 \cdot 7\) give \(2 \cdot 3 \cdot 7 = 42\).
- Check that both original numbers divide your answer with no remainder.
Least Common Multiple: solved example
Problem
Two shuttle buses leave a campus stop at 8:00 a.m. One returns every 12 minutes and the other every 18 minutes. When do they next leave together?
Solution
Multiples of 12: 12, 24, 36, 48. Multiples of 18: 18, 36, 54. The first match is 36.
Check with primes: \(12 = 2^2 \cdot 3\) and \(18 = 2 \cdot 3^2\), so the LCM is \(2^2 \cdot 3^2 = 36\). The buses leave together again after 36 minutes, at 8:36 a.m.
Try it
Question 1
Find the LCM of 8 and 20.
Show the answer
Count by 20s: 20 is not a multiple of 8, but 40 is (\(8 \cdot 5\)). So the answer is 40.
Question 2
Add \(\frac{1}{6} + \frac{3}{10}\) using the LCM of the denominators.
Show the answer
The LCM of 6 and 10 is 30. Then \(\frac{5}{30} + \frac{9}{30} = \frac{14}{30} = \frac{7}{15}\).
Least Common Multiple vs. Greatest Common Factor
The least common multiple grows outward: it is at least as large as the bigger number and comes from skip counting. The greatest common factor shrinks inward: it is at most as large as the smaller number and comes from splitting. Use the first for “when again”, the second for “largest equal groups”.
Keep going
Lessons
Worksheets and quizzes
- Greatest Common Factor and Least Common Multiple – Worksheet
- Greatest Common Factor and Least Common Multiple – Quiz
Related words
Whole dictionary: A–Z · 6th Grade · Numbers and Operations