Difference between revisions of "Least common multiple"

(Added examples, method for finding)
Line 1: Line 1:
The least common multiple (LCM) of two or more numbers is, obviously, the lowest multiple common to both.  Any set of numbers have an infinite amount of common [[multiples]] but only one LCM.
+
The "least common multiple" (LCM) of two or more numbers is the lowest multiple common to both.  Any set of numbers have an infinite amount of common [[multiples]] but only one LCM.
The most primitive way to find the LCM of a set of numbers is to list out the multiples of each until you find a multiple that is common to all of them. This is usually only used when the numbers are small.
+
 
 +
== How to find ==
 +
The most primitive way to find the LCM of a set of numbers is to list out the multiples of each until you find a multiple that is common to all of them. This is a tedious method, so it is usually only used when the numbers are small. For example, suppose we wanted to find the LCM of two numbers 4 and 6. We would begin by listing the multiples of 4 and 6 until we find the smallest number in both lists, as shown below.
 +
 
 +
4  8  '''12'''
 +
 
 +
6  '''12'''
 +
 
 +
12 is the LCM of 4 and 6.
 +
 
 +
The LCM of two numbers can be more easily be found by first finding the [[Greatest_common_divisor | greatest common divisor (GCD)]]. Once the GCD is known the LCM is calculated by the following equation, <math> LCM(a, b) = \frac{a \times b}{GCD(a, b)} </math>.
 +
 
 +
Let's use the above example again. The GCD of 4 and 6 is 2, since <math> 4 = 2^2 </math> and <math> 6 = 2 \times 3</math>. Using the above equation, we find <math> LCM(4, 6) = \frac{4 \times 6}{2} = \frac{24}{2} = 12 </math>.

Revision as of 08:59, 20 June 2006

The "least common multiple" (LCM) of two or more numbers is the lowest multiple common to both. Any set of numbers have an infinite amount of common multiples but only one LCM.

How to find

The most primitive way to find the LCM of a set of numbers is to list out the multiples of each until you find a multiple that is common to all of them. This is a tedious method, so it is usually only used when the numbers are small. For example, suppose we wanted to find the LCM of two numbers 4 and 6. We would begin by listing the multiples of 4 and 6 until we find the smallest number in both lists, as shown below.

4 8 12

6 12

12 is the LCM of 4 and 6.

The LCM of two numbers can be more easily be found by first finding the greatest common divisor (GCD). Once the GCD is known the LCM is calculated by the following equation, $LCM(a, b) = \frac{a \times b}{GCD(a, b)}$.

Let's use the above example again. The GCD of 4 and 6 is 2, since $4 = 2^2$ and $6 = 2 \times 3$. Using the above equation, we find $LCM(4, 6) = \frac{4 \times 6}{2} = \frac{24}{2} = 12$.

Invalid username
Login to AoPS