The smallest positive number that is a multiple of two or more numbers. Show
Let's start with an Example ...
List the Multiples of each number, The multiples of 3 are 3, 6, 9, 12, 15, 18, ... etc
Find the first Common (same) value: The Least Common Multiple of 3 and 5 is 15 (15 is a multiple of both 3 and 5, and is the smallest number like that.) So ... what is a "Multiple" ?We get a multiple of a number when we multiply it by another number. Such as multiplying by 1, 2, 3, 4, 5, etc, but not zero. Just like the multiplication table. Here are some examples:
What is a "Common Multiple" ?Say we have listed the first few multiples of 4 and 5: the common multiples are those that are found in both lists:
Notice that 20 and 40 appear in both lists? What is the "Least Common Multiple" ?It is simply the smallest of the common multiples. In our previous example, the smallest of the common multiples is 20 ... ... so the Least Common Multiple of 4 and 5 is 20. Finding the Least Common MultipleList the multiples of the numbers until we get our first match.
The multiples of 4 are: 4, 8, 12, 16, 20, ... Aha! there is a match at 20. It looks like this: So the least common multiple of 4 and 10 is 20
The multiples of 6 are: 6, 12, 18, 24, 30, ... There is a match at 30 So the least common multiple of 6 and 15 is 30 More than 2 NumbersWe can also find the least common multiple of three (or more) numbers.
Multiples of 4 are: 4, 8, 12, 16, 20, 24, 28, 32, 36, ... So 24 is the least common multiple (I can't find a smaller one!) Hint: We can have smaller lists for the bigger numbers. Least Common Multiple ToolThere is another method: the Least Common Multiple Tool does it automatically. (Yes, we waited until the end to tell you!) Copyright © 2018 MathsIsFun.com
The Least Common Multiple (LCM) is also referred to as the Lowest Common Multiple (LCM) and Least Common Divisor (LCD). For two integers a and b, denoted LCM(a,b), the LCM is the smallest positive integer that is evenly divisible by both a and b. For example, LCM(2,3) = 6 and LCM(6,10) = 30. The LCM of two or more numbers is the smallest number that is evenly divisible by all numbers in the set. Least Common Multiple CalculatorFind the LCM of a set of numbers with this calculator which also shows the steps and how to do the work. Input the numbers you want to find the LCM for. You can use commas or spaces to separate your numbers. But do not use commas within your numbers. For example, enter 2500, 1000 and not 2,500, 1,000. How to Find the Least Common Multiple LCMThis LCM calculator with steps finds the LCM and shows the work using 6 different methods:
How to Find LCM by Listing Multiples
Example: LCM(6,7,21)
How to find LCM by Prime Factorization
The LCM(a,b) is calculated by finding the prime factorization of both a and b. Use the same process for the LCM of more than 2 numbers. For example, for LCM(12,30) we find:
For example, for LCM(24,300) we find:
How to find LCM by Prime Factorization using Exponents
Example: LCM(12,18,30)
Example: LCM(24,300)
How to Find LCM Using the Cake Method (Ladder Method)The cake method uses division to find the LCM of a set of numbers. People use the cake or ladder method as the fastest and easiest way to find the LCM because it is simple division. The cake method is the same as the ladder method, the box method, the factor box method and the grid method of shortcuts to find the LCM. The boxes and grids might look a little different, but they all use division by primes to find LCM. Find the LCM(10, 12, 15, 75)
How to Find the LCM Using the Division MethodFind the LCM(10, 18, 25)
How to Find LCM by GCFThe formula to find the LCM using the Greatest Common Factor GCF of a set of numbers is: LCM(a,b) = (a×b)/GCF(a,b) Example: Find LCM(6,10)
A factor is a number that results when you can evenly divide one number by another. In this sense, a factor is also known as a divisor. The greatest common factor of two or more numbers is the largest number shared by all the factors. The greatest common factor GCF is the same as:
How to Find the LCM Using Venn DiagramsVenn diagrams are drawn as overlapping circles. They are used to show common elements, or intersections, between 2 or more objects. In using Venn diagrams to find the LCM, prime factors of each number, we call the groups, are distributed among overlapping circles to show the intersections of the groups. Once the Venn diagram is completed you can find the LCM by finding the union of the elements shown in the diagram groups and multiplying them together. How to Find LCM of Decimal Numbers
Properties of LCMThe LCM is associative:LCM(a, b) = LCM(b, a) The LCM is commutative:LCM(a, b, c) = LCM(LCM(a, b), c) = LCM(a, LCM(b, c)) The LCM is distributive:LCM(da, db, dc) = dLCM(a, b, c) The LCM is related to the greatest common factor (GCF):LCM(a,b) = a × b / GCF(a,b) and GCF(a,b) = a × b / LCM(a,b) References[1] Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae, 31st Edition, New York, NY: CRC Press, 2003 p. 101. [2] Weisstein, Eric W. Least Common Multiple. From MathWorld--A Wolfram Web Resource. |