Least Common Multiple

Pronunciation: /list ˈkɒm.ən ˈmʌl.tə.pəl/ Explain

A least common multiple is the expression of least degree or the least integer that is evenly divisible by two or more expressions or integers. Take the integers 2 and 5. The least integer that is evenly divisible by both 2 and 5 is 10. So 10 is the least common multiple of 2 and 5.

Number line with the multiples of 2 and 5 marked. The first common multiple is 10.

How to find a least common multiple

Example 1 - Find the least common multiple of 3 and 6.
StepEquationDescription
13 = 3 Find the prime factorization of the first integer.
22 · 3 = 6 Find all the prime factors of the second integer.
3The prime factors of 3 =3 and the prime factors of 6 = 2*3 are next to each other. An arrow connects the two threes and points to the 3 in the result. Another arrow points from the 2 to the 2 in the result. Combine the prime factors of the two integers.
Table 1: Least common multiple of 3 and 6.

Example 2 - Find the least common multiple of 10 and 12.
StepEquationDescription
110 = 2 · 5 Find all the prime factors of the first integer.
212 = 22 · 3 Find all the prime factors of the second integer.
3The prime factors of 10 = 2*5 and the prime factors of 12 = (2^2)*3 are next to each other. An arrow connects one set of twos with a two on the other side. An arrow points from the other two to the other side. A third arrows points from the 3 to a three on the other side. Combine the prime factors of the two Multiples.
Table 2: Least common multiple of 10 and 12.

The least common multiples of the denominators are used when adding or subtracting fractions. In this case they are called least common denominators.

Example 3: Find the least common multiple of x2 - x - 6 and x2 + 6x + 8
StepEquationDescription
1x2 - x - 6 = (x + 2)(x - 3) Find all the prime factors of the first expression.
2x2 + 6x + 8 = (x + 2)(x + 4) Find all the prime factors of the second integer.
3Arrows showing combining (x+2) and (x+2) and copying (x-3) and (x+4) Combine the prime factors of the two Multiples.
Table 3: Least common multiple of x2 - x - 6 and x2 + 6x + 8

References

  1. McAdams, David E.. All Math Words Dictionary, least common multiple. 2nd Classroom edition 20150108-4799968. pg 107. Life is a Story Problem LLC. January 8, 2015. Buy the book

Cite this article as:

 McAdams, David E. Least Common Multiple. 4/24/2019. All Math Words Encyclopedia. Life is a Story Problem LLC. https://www.allmathwords.org/en/l/leastcommonmultiple.html.

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Revision History

4/24/2019: Changed equations and expressions to new format. (McAdams, David E.)
12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)
8/31/2018: Removed broken links, updated license, implemented new markup. (McAdams, David E.)
9/30/2010: Added graphics to improve readability. (McAdams, David E.)
9/8/2010: Initial version. (McAdams, David E.)

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