In this article we learn “how to calculate lcm using prime factorization“. Numbers are factorized to find the least common multiple by the factorization method. We will consider this method as set by set.

First Step – We do the prime factors of the given numbers.

Example – If we have to find the least common denominator of 7, 14, 24. So first we will find their prime factors.

Prime prime factors of 8?

The prime factor of 8 is 2 × 2 × 2

Prime prime factors of 16?

Prime factorization of 16 = 2 x 2 x 2 x 2

Prime factorization of 24?

Prime factorization of 24 = 2x2x2x3 Second Step – In this step we organize the number. Lets keep small numbers first.

Prime factorization of 8 = 2 x2x2
Prime factorization of 16 = 2x2x2x2
Prime factorization of 24 = 2x2x2x3

Third Step – After arranging all the numbers the common numbers are chosen.

Prime factorization of 8 = 2 x2x2
Prime factorization of 16 = 2x2x2x2
Prime factorization of 24 = 2x2x2x3

Hence the common number is 2x2x2. Fourth Step – After selecting common numbers, we include the remaining numbers.

Prime factorization of 8 = 2 x2x2
Prime factorization of 16 = 2x2x2x2
Prime factorization of 24 = 2x2x2x3

Hence the remaining number is 2×3.

The product of all numbers and the remaining numbers is the least common multiple of 8, 16, 24.
The least common multiple of 8, 16, 24 = 48.

In this method, the least common divisor is found by dividing the numbers.

Example – Find the least common divisor of 6, 6, 12 using division method.

2|6, 8, 12 2|3, 4, 6 3|3,2,3
0|1,2,1
The least common multiple of 6, 8, 12 = 2ⅹ2ⅹ2ⅹ3 = 24
Note – In this method, the first is divided by small numbers.

The least common denominator is the smallest number that is completely divisible by the given numbers. Which is the definition of LCM (least common denominator)

Example

For example, the least common multiple of 6, 8,12 is 24.This means that 24 is the smallest number that is divisible by 6 8 12.

Definition Of LCM No-2

The least common denominator is the product of the smallest expressions. Which is completely divided by the given expressions.

Example

If the least common multiple of x2, x3, x4 is x5. So this means that x5 is the smallest expression which is completely divided by x2, x3, x4.

Definition Of LCM No-3

The smallest product of a number by two or by two which is completely divided by those numbers.

Example
The least common multiple of 3,612?
Table of 3 = 3, 4, 6, 12, 15, 14, 21, 26, 24,30.
Table of 6 = 6, 12 18, 24, 30, 36, 42, 48, 60.
Table of 12 = 12, 24,36, 48, 60, 72, 84, 96,108, 120.

We see that only 12 common are being found in all three numbers. We can say that the number is the least common multiple of 3,6, 12. Note – If the common number is two or more, then the smallest common number will be considered as the least common denominator.

Definition Of LCM No-4

We can understand the least common denominator in this context. Suppose there are three ball A, B, C. Whose diameters are 2, 9, 12. Which has two holes E and F (Size 12, 26). If the n ball is inserted into the hole, all three balls will be easily crossed. But we can call the smallest hole as the least common denominator.

Definition Of LCM No-5 With Help Of Venn Diagram

The above diagram shows two Venn diagrams. There is 24 in the middle of one and 4 in the middle of the other. And around them are 2, 4, 4, 7 which divide both the middle numbers. 24 is the smallest number. Which is divided by numbers 2, 4, 6, 4. This means that there cannot be a number smaller than 24 which can be divided by numbers 2, 4, 6, 7.

In this article we explain LCM And HCF Definition, Formula, Relation between lcm and hcf, questions

LCM And HCF Definition

LCM Definition ( Least Common Multiple )

The least common multiple of two or more numbers or expressions is the smallest number Or is an expression with a minimum of a square that is completely divisible by the given numbers or expressions.

Examples
Finding the least common multiple of 6, 8, 12 numbers.

The numbers divided by 6 are 6, 12, 18, 24, 30, 36

The numbers 8, 16, 24, 32, 40, divided by 8

The numbers 8, 16, 24, 32, 40, divided by 8

Numbers divided by 12 are 12, 24, 36

24 is the least common number among the numbers divided by these three numbers, so the least common multiple of these numbers is 24.

Similarly, the least common multiple of xЗ, x２, x５, x４ is x５.

Multiples of xЗ =, x４, x５, x６

Multiples of x２ = x２, xЗ, x４, x５, x６

Multiples of x５ = x５, x６, x７

Multiples of x４ = x４, x５, x６, x７

The common expression with the minimum address = x５. Thus, the least common multiple = x５

Method Of Finding LCM ( Least Common Multiple )

There are two methods to find the LCM .

Factor Method

In this method we first do a prime factorization of numbers.

Then we get the product of each factor of more powers.

For example, to find the least common multiple of 6,8,12, one must first do a prime factorization of these numbers.

Prime factorization of 6 = 2 × 3

Prime factorization of 12 = 2 × 2 × 3

Prime factorization of 8 = 2 × 2 × 2

Note – When we find the least common multiple, we include the common numbers as well as the other remaining numbers.

Explanation
Step 1 — The prime factor of 6,12,8 will include the common number 2. The following is the picture.

Step-2 Common will take 2 out of the prime factors of 8,12. In the following figure.

Step-3 Common will take 3 out of the prime factors of 6,12. In the following figure.

Step-4 Finally, the number 2 in the prime factor of 12 will also be included. In the following figure.

Thus, after following all the steps, the least common multiple of the number 6,8,12 = 2 x 2 x 3 x 2 = 24

Example-2
Find the least common multiple of xy, x２y, xy２?

Solve– The least common multiple of x 2 y = x❌x ❌y.

The least common multiple of x y2 = x❌y ❌y

The least common multiple of x y = x❌y

x y, x 2 y, the least common multiple of x y2 = x 2 y2
0000000000000 Explanation Step-1 Common number in the factors of x y, x 2 y, x y2 = X . In the picture.

Step-2 Common number in the factors of x y, x 2 y, x y2 =Y . In the picture.

Step-3 At the end, the remaining XY will be included in the factors of x 2 y, x y2.

Step-4

Division Method

In this method, the least common divisor is obtained by dividing by prime numbers.

Example

To find the least common multiple of 6, 8, 12 by this method.

Minimum common factor of 6,8,12 = 2 x 2 x 3 x 1 x 2 x 1 = 24

Highest Common Factor

Defination of Highest Common Factor

The largest number that completely divides each of the given numbers.The numbers are called the greatest common factor.

Example

Finding the maximum common factor of 6,8,12?

The numbers dividing 6 = 1,2,3,6.

The numbers dividing 8 = 1,2,4,8.

The number dividing 12 is = 1,2,3,4,6,12.

The maximum common number is 2 among the numbers dividing 8,14,12. It is clear that their maximum common factor is 2.

Method of finding HCF

1. Factor Method

In this method, the prime factors of the given numbers are derived.Then the product of all the rhinoceros is the greatest common factor.

Example

Finding the maximum common factor of 6,8,12?

6 = 2×8

8 = 2x2x2

12 = 2x2x2x3

than maximum common factor is 2

2. Division Method

In this method, the first is divided by the smallest number.