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# Lowest Common Multiple

In Mathematics, the Lowest Common Multiple (LCM) of any two or more natural numbers is the number that is the lowest of their common multiples. It is also called Least Common Multiple or Least Common Divisor (LCD). For example, the lowest common multiple of 4 and 6 is 12. Therefore, 12 is the smallest common multiple such that 4 and 6 are the factors of 12.

Now by the definition of factors of a number, we know that a factor divides the original number, exactly. So by this we can conclude that the lowest common multiple of two or more numbers is evenly divisible by the given numbers. Thus, the numbers are the factors of their lowest common multiple.

Let us learn to find the LCM of given numbers with the help of examples, here at BYJU’S. This will help students of Class 6 to solve problems based on Least common factors. There are problems related to addition and subtraction of fractions that use the concept of LCM to make the denominators the same. Hence, it is necessary to have a core knowledge of Lowest common multiple, to solve such arithmetic problems.

## How to Find the Lowest Common Multiple?

To find the lowest common multiple (LCM), we can use these given methods:

• Prime factorisation method
• Listing multiples

Both these methods are very simple and easy to understand.

### Prime Factorisation Method

In the prime factorisation method, the LCM of the two or more numbers is the product of the prime factors counted the maximum number of times they appear in any of the numbers. Let us see a few examples here.

### Solved Examples

Example 1: What is the LCM of 14 and 18?

Solution: By prime factorisation, we can write,

14 = 2 x 7

18 = 2 x 3 x 3

The prime factors 2, 3, 3 and 7 are the maximum number of times they occurred in the numbers. So, product of these prime factors will result in required LCM. Therefore,

LCM of 14 and 18 = 2 x 3 x 3 x 7 = 126

LCM (14, 18) = 126

Example 2: Find the LCM of 32 and 90.

Solution: By prime factorisation of 32 and 90, we get;

32 → 2 x 2 x 2 x 2 x 2 = 25

90 → 2 x 3 x 3 x 5 = 2 x 32 x 5

The new list of factors will be 2, 2, 2, 2, 2, 3, 3, 5, that have occurred at maximum times in the numbers. So, multiplication of all these factors will result in required LCM.

Hence, LCM of 32 and 90 will be;

LCM (32, 90) = 2 x 2 x 2 x 2 x 2 x 3 x 3 x 5 = 1440

Example 3: Find the LCM of 35, 42, 50.

Solution: By prime factorisation method we have;

35 → 5 x 7

42 → 2 x 3 x 7

50 → 2 x 5 x 5

The list of factors that have occurred the maximum number of times are 2, 3, 5, 5, 7. So, the product of these factors will be the required LCM.

LCM (35, 42, 50) = 2 x 3 x 5 x 5 x 7 = 1050

### Listing Multiples

In this method, we list down the multiples of each of the numbers given and then find the lowest common multiple among them. Listing out the first five multiples of the numbers will be enough sometimes to get the required LCM. But in a few cases, we need to list the multiples of the numbers as far as we get the common multiple. Let us understand with examples.

### Solved Examples

Example 1: Find the LCM of 8 and 12 using the listing method.

Solution: Let us first list down the multiples of the given numbers.

Multiples of 8 = 8, 16, 24, 32, 40

Multiples of 12 = 12, 24, 36, 48

As we can see, the first common multiple is 24.

Therefore, LCM of 8 and 12 is 24.

Example 2: Find the LCM of 10, 12, 15 using listing methods.

Solution: First listing all the multiples, we get;

Multiples of 10 = 10, 20, 30, 40, 50, 60, 70, 80

Multiples of 12 = 12, 24, 36, 48, 60, 72, 84

Multiples of 15 = 15, 30, 45, 60, 75, 90

Therefore, LCM of (10, 12, 15) = 60.

## LCM by Division Method

• In this method, we divide the given numbers by the least prime number which divides at least one of the given numbers.
• The numbers that are not divisible by the prime number are written as it is in the next row.
• Now, again we repeat the division process by dividing the next row numbers by prime numbers
• We repeat the division unless we get all 1 in the last row.

Let us solve an example, to understand this method.

Example: Find the LCM of 30, 35 and 40.

Solution: Using the division method, we have;

 2 30 35 40 2 15 35 20 2 15 35 10 3 15 35 5 5 5 35 5 7 1 7 1 x 1 1 1

Hence, LCM (30, 35, 40) = 2 x 2 x 2 x 3 x 5 x 7 = 840

## Practice Questions

Find the LCM of the following number:

• 15 and 60
• 10, 20 and 25
• 12, 16, 24 and 36
• 75 and, 69
• 15 and 4
• 9, 15 and 45

## Frequently Asked Questions on Lowest Common Multiple

### What is Lowest common multiple? Give an example.

Lowest Common Multiple (LCM) of any two or more natural numbers is the number that is the lowest of their common multiples. For example, LCM of 3 and 7 is 21.

### How to calculate the lowest common multiple numbers?

We can use the prime factorisation method or listing the multiples of given numbers, to find the LCM. Also, even division by prime numbers is a method to find the LCM, unless all the numbers are divided completely.

### How are HCF and LCM related?

The relation between LCM and HCF is given by:
LCM (a,b) = axb/HCF(a,b)
Where a and b are two different numbers

### What is the LCM of 6 and 12?

The LCM of 6 and 12 is 12.

### What is the LCM of 24 and 36?

The LCM of 24 and 36 is 72.