# Difference between revisions of "Least common multiple"

(→How to find) |
|||

(3 intermediate revisions by 2 users not shown) | |||

Line 1: | Line 1: | ||

− | The '''least common multiple''' (LCM) of two or more [[positive integer]]s is the smallest [[integer]] which is a [[multiple]] of all of them. Any [[finite]] [[set]] of integers has an [[infinite]] number of common | + | The '''least common multiple''' ('''LCM''') of two or more [[positive integer]]s is the smallest [[integer]] which is a [[multiple]] of all of them. Any [[finite]] [[set]] of integers has an [[infinite]] number of [[common multiple]]s, but only one LCM. The LCM of a set of numbers <math>\{a_1,a_2,\cdots,a_n\}</math> is conventionally represented as <math>[a_1,a_2,\ldots,a_n]</math>. |

+ | |||

+ | == Video on Least Common Multiple == | ||

+ | |||

+ | [https://youtu.be/1lfwMN1cUwY Least Common Multiple] | ||

== How to find == | == How to find == | ||

− | ===Brute force=== | + | === Brute force === |

Line 10: | Line 14: | ||

12 is the LCM of 4 and 6. | 12 is the LCM of 4 and 6. | ||

− | ===Using prime factorization=== | + | === Using prime factorization === |

The LCM of two or more numbers can also be found using [[prime factorization]]. In order to do this, factor all of the numbers involved. For each [[prime number]] which divides any of them, take the largest [[exponentiation | power]] with which it appears, and multiply the results together. For example, to find the LCM of 8, 12 and 15, write: | The LCM of two or more numbers can also be found using [[prime factorization]]. In order to do this, factor all of the numbers involved. For each [[prime number]] which divides any of them, take the largest [[exponentiation | power]] with which it appears, and multiply the results together. For example, to find the LCM of 8, 12 and 15, write: | ||

Line 21: | Line 25: | ||

Three primes appear in these factorizations, 2, 3 and 5. The largest power of 2 that appears is <math>2^3</math>; the largest power of 3 that appears is <math>3^1</math>; and the largest power of 5 that appears is <math>5^1</math>. Therefore the LCM, <math>LCM(8, 12, 15) = 2^3\cdot 3^1\cdot 5^1 = 120</math>. | Three primes appear in these factorizations, 2, 3 and 5. The largest power of 2 that appears is <math>2^3</math>; the largest power of 3 that appears is <math>3^1</math>; and the largest power of 5 that appears is <math>5^1</math>. Therefore the LCM, <math>LCM(8, 12, 15) = 2^3\cdot 3^1\cdot 5^1 = 120</math>. | ||

− | ===Using the GCD=== | + | === Using the GCD === |

The LCM of two numbers can be found more easily by first finding their [[greatest common divisor]] (GCD). Once the GCD is known, the LCM is calculated by the following equation, <math> LCM(a, b) = \frac{a \cdot b}{GCD(a, b)} </math>. | The LCM of two numbers can be found more easily by first finding their [[greatest common divisor]] (GCD). Once the GCD is known, the LCM is calculated by the following equation, <math> LCM(a, b) = \frac{a \cdot b}{GCD(a, b)} </math>. | ||

Let's use our first example. The GCD of 4 and 6 is 2. Using the above equation, we find <math> LCM(4, 6) = \frac{4 \cdot 6}{2} = \frac{24}{2} = 12 </math>, just like we expected. | Let's use our first example. The GCD of 4 and 6 is 2. Using the above equation, we find <math> LCM(4, 6) = \frac{4 \cdot 6}{2} = \frac{24}{2} = 12 </math>, just like we expected. | ||

+ | |||

+ | [[Category:Definition]] | ||

+ | [[Category:Number theory]] |

## Latest revision as of 14:21, 24 December 2020

The **least common multiple** (**LCM**) of two or more positive integers is the smallest integer which is a multiple of all of them. Any finite set of integers has an infinite number of common multiples, but only one LCM. The LCM of a set of numbers is conventionally represented as .

## Contents

## Video on Least Common Multiple

## How to find

### Brute force

4 8 **12**

6 **12**

12 is the LCM of 4 and 6.

### Using prime factorization

The LCM of two or more numbers can also be found using prime factorization. In order to do this, factor all of the numbers involved. For each prime number which divides any of them, take the largest power with which it appears, and multiply the results together. For example, to find the LCM of 8, 12 and 15, write:

Three primes appear in these factorizations, 2, 3 and 5. The largest power of 2 that appears is ; the largest power of 3 that appears is ; and the largest power of 5 that appears is . Therefore the LCM, .

### Using the GCD

The LCM of two numbers can be found more easily by first finding their greatest common divisor (GCD). Once the GCD is known, the LCM is calculated by the following equation, .

Let's use our first example. The GCD of 4 and 6 is 2. Using the above equation, we find , just like we expected.