## What is the Least common factor (LCM)?

The least common multiple or lowest common multiples is the least number which is divisible by the numbers for which you are finding the LCM.

**For example:**

The L.C.M of **3, 5** and **6** is **30**.

This means that the **30** is the common number that can be divided by any of the numbers between **3,5** and **6** to get the answer in integers.

## Methods of finding the LCM

There are multiple methods of finding the Lowest common multiple between multiple numbers. Some of the ways are discussed below:

- Listing Multiples
- Prime Factorization
- Prime Factorization using exponents
- Cake Method (also known as ladder method)
- Division Method

## How to find the LCM manually?

As discussed above, there are several methods to find the LCM. Let’s go through the important ones to understand the basic idea.

### 1. Prime Factorization

Prime factorization is a simple method to find the LCM. Let’s see how to find LCM using prime factorization step by step.

#### Example:

Find LCM of **400, 500,** and **550** with **prime factorization?**

Solution:

Step 1: List the factors of the given numbers.**400:** 5 × 5 × 2 × 2 × 2 × 2 = 5^{2} × 2^{4}**500:** 5 × 5 × 5 × 2 × 2 = 5^{3} × 2 ^{2}**550:** 5 × 5 × 2 × 11 = 5^{2} × 2 × 11

Step 2: Multiply the prime factors to get the LCM. If there are common factors, use them only once.

5^{3} × 2^{4} × 11 = **22000**

So, the least common multiple of **400, 500**, and **550** is **22000.**

### 2. Division Method

The division method is another method to get the LCM of multiple numbers. Follow the below steps to obtain the least common multiple using the division method.

#### Example:

Find the least common multiple of **40** and **50** with the **division method?**

Step 1: Place all numbers on a horizontal line as shown in the image below.

Step 2: Start dividing the numbers with the least prime number. Write the quotient of division exactly below the dividend. Keep dividing until all numbers are fully divided and the remainder is 1.

Step 3: Multiply all of the **divisors** to get the LCM.**2 × 2 × 2 × 5 × 5 = 200**

So, the LCM of **40** and **50** is **200** using the division method.

### 3. Listing Multiples

The simplest way to find the LCM of any numbers is to find through the listing multiples. In this method, you have to list down all the multiple of the numbers and then find the common multiple between these numbers which would be the LCM.

#### Example:

Find the LCM of **8** and **14** using **lists of multiple method?**

Step 1: Write down the multiples of the given numbers.

Multiples of **8 = 8, 16, 24, 32, 40, 48, 56, 64, 72**

Multiples of **14 = 14, 28, 42, 56, 70, 84**

Step 2: Look out for the common multiple in the multiples of all numbers.

Multiples of **8 = 8, 16, 24, 32, 40, 48, 56, 64, 72**

Multiples of **14 = 14, 28, 42, 56, 70, 84**

As we can see that the common number between the multiples of **8** and **14** is **56**, which is the least common factor for these numbers.

LCM of **8** and **14** = **56**

## Properties of LCM

- It is associative
- It is commutative
- It is distributive

## LCM Formula

#### Finding the LCM through GCF

Finding the least common multiple with the value of the greatest common factor is possible with the following formula:`LCM(a,b) = (a x b) / GCF(a,b)`

**Example:**

The LCM of **(8,14)****GCF(8,14) = 2**

Using formula: **(8 x 14) / 2****LCM = 56**

## LCM Examples

Questions | Answer | Action |
---|---|---|

lcm of 9 and 12 | 36 | Convert |

lcm of 8 and 14 | 56 | Convert |

lcm of 9 and 15 | 45 | Convert |

lcm of 8 and 12 | 24 | Convert |

lcm of 6 and 9 | 18 | Convert |

lcm of 12 and 15 | 60 | Convert |

lcm of 8 and 10 | 40 | Convert |

lcm of 4 and 6 | 12 | Convert |

lcm of 10 and 12 | 60 | Convert |

lcm of 6 and 12 | 12 | Convert |

lcm of 15 and 20 | 60 | Convert |

lcm of 12 and 18 | 36 | Convert |

lcm of 16 and 20 | 80 | Convert |

lcm of 24 and 30 | 120 | Convert |

lcm of 8,12 and 16 | 48 | Convert |

lcm of 9,10 and 28 | 1260 | Convert |