In mathematics computation of the least common multiple and greatest common divisors of two or more numbers. LCM is the smallest integer which is a multiple of two or more numbers. For example, LCM of 4 and 6 is 12, and LCM of 10 and 15 is 30. As with the greatest common divisors, there are many methods for computing the least common multiples also. One method is to factor both numbers into their primes. The LCM is the product of all primes that are common to all numbers. In this topic, we will discuss the concept of least common multiple and LCM formula with examples. Let us learn it!

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**LCM Formula**

**What is LCM?**

The Least Common Multiple i.e. LCM of two integers a and b is that smallest positive integer which is divisible by both a and b. Thus the smallest positive number is a multiple of two or more numbers.

For example, to calculate lcm of (40, 45), we will find factors of 40 and 45, getting

40 is expressed as 2× 2 × 2 × 5

45 is expressed as 3 × 3 × 5

The prime factors common to one or the other are 2, 2, 2, 3, 3, 5.

Thus the least common multiple will be 2 × 2 × 2 × 3 × 3 × 5 = 360.

**To find out LCM using prime factorization method:**

Step 1: Show each number as a product of their prime factors.

Step 2: LCM will be the product of the highest powers of all prime factors.

**To find out the LCM using division Method:**

Step 1: First, we need to write the given numbers in a horizontal line separated by commas.

Step 2: Then, we need to divide all the given numbers by the smallest prime number.

Step 3: We now need to write the quotients and undivided numbers in a new line below the previous one.

Step 4: Repeat this process until we find a stage where no prime factor is common.

Step 5: LCM will be the product of all the divisors and the numbers in the last line.

**L.C.M formula for any two numbers:**

1) For two given numbers if we know their greatest common divisor i.e. GCD, then LCM can be calculated easily with the help of given formula:

2) To get the LCM of two Fractions, then first we need to compute the LCM of Numerators and HCF of the Denominators. Further, both these results will be expressed as a fraction. Thus,

**Solved Examples**

Q.1: Find out the LCM of 8 and 14.

Solution:

Step 1: First write down each number as a product of prime factors.

8 = 2× 2 × 2 = 2³

14 = 2 × 7

Step 2: Product of highest powers of all prime factors.

Here the prime factors are 2 and 7

The highest power of 2 here = 2³

The highest power of 7 here = 7

Hence LCM = 2³ × 7 = 56

Q.2: If two numbers 12 and 30 are given. HCF of these two is 6 then find their LCM.

Solution: We will use the simple formula of LCM and GCD.

a = 12

b =30

gcd =6

Thus

## Least Common Multiple Formula

When two or more numbers have the same numbers as their multiples, these multiples are called common multiples. Out of all the common possible multiples the one that is least (smallest) and common for both the numbers, is said to be their least common multiple LCM. LCM Formula helps in calculating this least common multiple for numbers.

In other words, LCM for some integers, say a and b is, the smallest positive integer which is divisible by both a and b.

## What Is the LCM Formula?

The LCM formula can be expressed as,

### LCM Formula:

LCM = (a × b)/HCF(a,b)

where,

- a and b = Two terms
- HCF(a, b) = Highest common multiple of a and b

## Examples Using Least Common Multiple

**Example 1:** Find the LCM of 14 and 35.

**Solution: **

To find: Least common multiple of 14 and 35

Given:

a = 14

b = 35

Prime factorization of 14 and 35 is given as,

14 = 2 × 7

35 = 5 × 7

Using least common multiple formula,

L.C.M = 2 × 5 × 7

= 70

**Answer:** L.C.M. of 14 and 35 is 70.

**Example 2:** Find the LCM of 42 and 56.

**Solution:**

To find: Least common multiple of 42 and 56

Given:

a = 42

b = 56

Prime factorization of 42 and 56:

42 = 2 × 3 × 7

56 = 2 × 2 × 2 × 7

Using least common multiple formula,

L.C.M = 2 × 2 × 2 × 3 × 7

= 168

**Answer: **L.C.M. of 14 and 35 is 168.

**To Find Out LCM Using Prime Factorization Method:**

Step 1: You need to show each number as a product of their prime factors.

Step 2: LCM will be the product of the highest powers of all prime factors.

Now we know what is LCM method, let’s solve a few examples.

**For Example, Let’s Find the LCM (12,30) We Find:**

First find the prime factorization of 12 = 2 × 2 × 3

Second, the prime factorization of 30 = 2 × 3 × 5

Using all prime numbers we have found as often as each occurs most often we take 2 × 2 × 3 × 5 = 60

Therefore, the LCM (12,30) = 60.

**To Find Out the LCM Using Division Method:**

Step 1: First, we need to write the given numbers in a horizontal line separated by commas.

Step 2: Then, we need to divide all the given numbers by the smallest prime number.

Step 3: We now need to write the quotients and undivided numbers in a new line below the previous one.

Step 4: Now you need to repeat this process until we find a stage where no prime factor is common.

Step 5: LCM will be the product of all the divisors and the numbers in the last line.

Now we know what is lcm method, let’s solve a few examples.

**For Example, Let’s Find the LCM (12,30) We Find:**

First find the prime factorization of 12 = 2 × 2 × 3

Second, the prime factorization of 30 = 2 × 3 × 5

Using all prime numbers we have found as often as each occurs most often we take 2 × 2 × 3 × 5 = 60

Therefore, the LCM (12,30) = 60.

**Least Common Multiple Formula for Any Two Numbers:**

1) For two given numbers if we know their greatest common divisor that is GCD, then the Least Common Multiple can be calculated easily using the help of given least common multiple formula:

2) To get the LCM of two Fractions, then first we need to compute the LCM of Numerators and HCF of the Denominators. Further, both these results will be expressed as a fraction. Thus,

**Questions to be Solved :**

**Question 1:** Find out the LCM of 8 and 14.

**Solution: **

Step 1: First you need to write down each number as a product of prime factors.

8 = 2× 2 × 2 = 2³

14 = 2 × 7

Step 2: Product of highest powers of all prime factors.

Here the prime factors are 2 and 7

The highest power of 2 here = 2³

The highest power of 7 here = 7

Hence LCM = 2³ × 7 = 56

**Question 2: Find the LCM(12,18,30).**

**Solution:** List down the prime factors of 12 = 2 × 2 × 3 = 2^{2} × 3^{1}

List down the prime factors of 18 = 2 × 3 × 3 = 2^{1} × 3^{2}

List down the prime factors of 30 = 2 × 3 × 5 = 2^{1} × 3^{1} × 5^{1}

You need to list all the prime numbers found, the number of times as they occur most often for any one given number and you need to multiply them together to find the Least common multiple.

After multiplying, 2 × 2 × 3 × 3 × 5 = 180

Using the concept of exponents instead, multiply together each of the prime numbers with the highest power

In exponential form, 2^{2} × 3^{2} × 5^{1} = 180

So, the LCM(12,18,30) = 180

## FAQs (Frequently Asked Questions)

**Question 1: What is the LCM of 8 and 12?**

Answer: The first multiple that 8 and 12 have in common is 24. We notice that 48 is also a common multiple; however, 24 is the smallest number that they have in common. This makes it the least common multiple.

**Question 2: What is LCM Example?**

Answer: The smallest positive number that is a multiple of two or more numbers. Example: the Least Common Multiple of 3 and 5 is equal to 15. Because the number15 is a multiple of number 3 and also a multiple of number 5 and it is the smallest number like that. 3 and 5 have other common multiples such as 30, 45, etc, but these numbers are all larger than 15.

**Question 3: What is LCM of Two Numbers?**

Answer: LCM (Least Common Multiple) of two numbers is the smallest number which can be divided by both numbers. For example LCM of 15 and 20 is 60 and LCM of the numbers 5 and 7 is equal to 35.

**Question 4: What is LCM Mean in Math?**

Answer: Least Common Multiple

LCM stands for Least Common Multiple. A multiple is a number you get when you multiply a number by a whole number (greater than 0). A factor is one of the numbers that multiplies by a whole number to get that number. example: the multiples of 8 are 8, 16, 24, 32, 40, 48, 56 the factors of 8 are 1, 2, 4, 8

## Calculator Use

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 Calculator

Find 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 LCM

This LCM calculator with steps finds the LCM and shows the work using 6 different methods:

- Listing Multiples
- Prime Factorization
- Cake/Ladder Method
- Division Method
- Using the Greatest Common Factor GCF
- Venn Diagram

## How to Find LCM by Listing Multiples

- List the multiples of each number until at least one of the multiples appears on all lists
- Find the smallest number that is on all of the lists
- This number is the LCM

Example: LCM(6,7,21)

- Multiples of 6: 6, 12, 18, 24, 30, 36,
**42**, 48, 54, 60 - Multiples of 7: 7, 14, 21, 28, 35,
**42**, 56, 63 - Multiples of 21: 21,
**42**, 63 - Find the smallest number that is on all of the lists. We have it in bold above.
- So LCM(6, 7, 21) is 42

## How to find LCM by Prime Factorization

- Find all the prime factors of each given number.
- List all the prime numbers found, as many times as they occur most often for any one given number.
- Multiply the list of prime factors together to find the LCM.

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:**

- Prime factorization of 12 = 2 × 2 × 3
- Prime factorization of 30 = 2 × 3 × 5
- Using all prime numbers found as often as each occurs most often we take 2 × 2 × 3 × 5 = 60
- Therefore LCM(12,30) = 60.

**For example, for LCM(24,300) we find:**

- Prime factorization of 24 = 2 × 2 × 2 × 3
- Prime factorization of 300 = 2 × 2 × 3 × 5 × 5
- Using all prime numbers found as often as each occurs most often we take 2 × 2 × 2 × 3 × 5 × 5 = 600
- Therefore LCM(24,300) = 600.

## How to find LCM by Prime Factorization using Exponents

- Find all the prime factors of each given number and write them in exponent form.
- List all the prime numbers found, using the highest exponent found for each.
- Multiply the list of prime factors with exponents together to find the LCM.

Example: LCM(12,18,30)

- Prime factors of 12 = 2 × 2 × 3 = 2
^{2}× 3^{1} - Prime factors of 18 = 2 × 3 × 3 = 2
^{1}× 3^{2} - Prime factors of 30 = 2 × 3 × 5 = 2
^{1}× 3^{1}× 5^{1} - List all the prime numbers found, as many times as they occur most often for any one given number and multiply them together to find the LCM
- 2 × 2 × 3 × 3 × 5 = 180

- Using exponents instead, multiply together each of the prime numbers with the highest power
- 2
^{2}× 3^{2}× 5^{1}= 180

- 2
- So LCM(12,18,30) = 180

Example: LCM(24,300)

- Prime factors of 24 = 2 × 2 × 2 × 3 = 2
^{3}× 3^{1} - Prime factors of 300 = 2 × 2 × 3 × 5 × 5 = 2
^{2}× 3^{1}× 5^{2} - List all the prime numbers found, as many times as they occur most often for any one given number and multiply them together to find the LCM
- 2 × 2 × 2 × 3 × 5 × 5 = 600

- Using exponents instead, multiply together each of the prime numbers with the highest power
- 2
^{3}× 3^{1}× 5^{2}= 600

- 2
- So LCM(24,300) = 600

## 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)

- Write down your numbers in a cake layer (row)

Divide the layer numbers by a prime number that is evenly divisible into two or more numbers in the layer and bring down the result into the next layer.

- The LCM is the product of the numbers in the L shape, left column and bottom row. 1 is ignored.
- LCM = 2 × 3 × 5 × 2 × 5
- LCM = 300
- Therefore, LCM(10, 12, 15, 75) = 300

## How to Find the LCM Using the Division Method

Find the LCM(10, 18, 25)

- Write down your numbers in a top table row

Starting with the lowest prime numbers, divide the row of numbers by a prime number that is evenly divisible into at least one of your numbers and bring down the result into the next table row.

- The LCM is the product of the prime numbers in the first column.
- LCM = 2 × 3 × 3 × 5 × 5
- LCM = 450
- Therefore, LCM(10, 18, 25) = 450

## How to Find LCM by GCF

The 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)

- Find the GCF(6,10) = 2
- Use the LCM by GCF formula to calculate (6×10)/2 = 60/2 = 30
- So LCM(6,10) = 30

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:

- HCF – Highest Common Factor
- GCD – Greatest Common Divisor
- HCD – Highest Common Divisor
- GCM – Greatest Common Measure
- HCM – Highest Common Measure

## How to Find the LCM Using Venn Diagrams

Venn 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

- Find the number with the most decimal places
- Count the number of decimal places in that number. Let’s call that number D.
- For each of your numbers move the decimal D places to the right. All numbers will become integers.
- Find the LCM of the set of integers
- For your LCM, move the decimal D places to the left. This is the LCM for your original set of decimal numbers.

## Properties of LCM

### The 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)

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