Key takeaway: The least common multiple (LCM) of two numbers is the first value that appears in both of their lists of multiples, for example 140 for 28 and 20.
Grade 66.NS.B.4Numbers and Operations
Definition of Least Common Multiple
The least common multiple, or LCM, is the first number that shows up in the multiplication lists of two numbers at the same time. Multiples of 28 are 28, 56, 84, 112, 140. Multiples of 20 are 20, 40, 60, 80, 100, 120, 140. The first match is 140.
The least common multiple answers “when do two repeating events line up again?” and supplies the denominator you need to add unlike fractions. With prime factorization, take every prime that appears, each to its highest power.
Formula and notation
The LCM of two numbers is written
\[\text{LCM}(28, 20) = 140\]
For two numbers, the product equals the GCF times the LCM: \(28 \times 20 = 560 = 4 \times 140\). That relationship gives a shortcut when the GCF is known.
Key facts
- Counting up from the bigger number never gives an LCM smaller than that number.
- If two numbers share no prime factor, their LCM is their product.
- Every common multiple is itself a multiple of the LCM.
- The least common multiple of two denominators is their least common denominator.
Least Common Multiple example
A harbor lighthouse blinks every 28 seconds, while a channel marker blinks every 20 seconds. Both blink at 9:00 p.m. When is the next moment both blink at once?
-
Factor: \(28 = 2^2 \times 7\) and \(20 = 2^2 \times 5\).
-
Take each prime at its highest power: \(2^2 \times 5 \times 7\).
-
Multiply: \(4 \times 5 \times 7 = 140\).
The lights meet again after 140 seconds, which is 2 minutes 20 seconds.
Common mistake
Watch out
Elias multiplies \(28 \times 20 = 560\) and calls that the LCM.
Why it is wrong
The product 560 does work as a shared multiple, yet it is not the first one, because 28 and 20 have the factor 4 in common. Dividing the product by that factor gives \(560 : 4 = 140\), which is smaller. Scan the lists of multiples for an earlier match.
Related terms and lessons
Browse the glossary: A–Z list · Grade 6 vocabulary · Numbers and Operations