How To Find The Lowest Common Factor

HCF and LCM

The highest mutual factor (HCF) of two or more given numbers is the largest number which divides each of the given numbers without leaving any residue. The everyman mutual multiple (LCM) of two or more than numbers is the smallest of the common multiples of those numbers. Information technology is very important to larn HCF and LCM in mathematics as it helps us to exercise our day-to-24-hour interval problems related to group and sharing. Let'southward learn about the dissimilar methods used to notice the HCF and LCM of numbers.

one.	What is HCF and LCM?
two.	How to Find HCF and LCM?
iii.	HCF and LCM Formula
4.	Difference between HCF and LCM
5.	FAQs on HCF and LCM

What is HCF and LCM?

HCF is defined as the highest common factor nowadays in two or more given numbers. It is as well termed equally the "Greatest Common Divisor" (GCD). For instance, the HCF of 24 and 36 is 12, because 12 is the largest number which can divide both the numbers completely. Similarly, the least common multiple (LCM) of two or more numbers is the smallest number which is a common multiple of the given numbers. For example, allow us take 2 numbers 8 and 16. Multiples of eight are: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, and so on. The multiples of sixteen are sixteen, 32, 48, 64, 80, 96, and so on. The offset common value among these multiples is the least mutual multiple (LCM) for eight and xvi, which is 16. Now, let u.s.a. learn two unremarkably used methods to find HCF and LCM.

How to Discover HCF and LCM?

There are various methods to find the HCF and LCM of numbers. The nigh common methods are:

Prime factorization method
Division method

Let us discuss these methods in item.

Finding HCF and LCM by Prime Factorization

Past using the prime factorization method for finding LCM and HCF, we start need to find the prime factors of the given numbers by using either the ladder method or the factor tree method. And so, nosotros can calculate the values of HCF and LCM by following the process explained below.

HCF by Prime number Factorization

To find the HCF of the given numbers by prime number factorization, we find the prime number factors of those numbers. After finding the factors, nosotros find the product of the prime number factors that are mutual to each of the given numbers. For example, let us find the HCF of 50 and 75 by the prime number factorization method.

The prime number factors of 50 = 2 × 5 × 5
The prime number factors of 75 = 3 × five × v

The mutual factors of l and 75 are 5 × 5. Thus, HCF of (50, 75) = 25.

LCM past Prime number Factorization

To calculate the LCM of any given numbers using the prime factorization method, we follow the steps given below:

Step one: List the prime number factors of the given numbers and annotation the common prime factors.
Step 2: The LCM of the given numbers = product of the common prime factors and the uncommon prime factors of the numbers.

Note: Common factors will be included only once.

Let us find the LCM of 160 and xc using prime factorization.

Footstep 1: The prime factors of 160 = 2× 2 × 2 × ii × two × 5 and xc = 2 × iii× iii × 5.
Step 2: The product of all the prime number factors = Common prime factors (ii × 5) × Uncommon prime factors (two × ii × two × 2 × 3 × 3) = 1440.

Therefore, LCM of 160 and 90 = 1440.

Finding HCF and LCM by Sectionalization Method

There are 2 unlike means to apply the sectionalization method to find LCM and HCF. Let us larn it 1 past one.

HCF by Partitioning Method

To find the HCF by division method, follow the steps given below:

Step 1: First, we need to divide the larger number by the smaller number and bank check the residual.
Stride 2: Make the remainder of the in a higher place step as the divisor and the divisor of the to a higher place step as the dividend and perform the division over again.
Step 3: Continue the division procedure till the residue is non equal to 0.
Stride iv: The last divisor will be the HCF of the given numbers.

Let's understand this method using an case. Here, we volition find HCF of 198 and 360 using the division method. Read out the following steps and chronicle them with the paradigm below.

Divide 360 past 198. The obtained residual is 162.
Make 162 equally the divisor and 198 as the dividend and perform the division again. Here the obtained balance is 36.
Brand 36 as the divisor and 162 equally the dividend and perform the sectionalization again. Here the obtained residual is 18.
Make 18 equally the divisor and 36 every bit the dividend and perform the division once again. Hither the obtained remainder is 0.
The last divisor, 18, is the HCF of 360 and 198.

Using division method to find HCF and LCM of two numbers
LCM by Partition Method

To find the LCM of numbers by the partitioning method, we divide the numbers with prime numbers and stop the division process when nosotros get only 1 in the terminal row. Observe the steps given below to observe the LCM of the given numbers using the division method.

Stride 1: Divide the numbers past the smallest prime number such that the prime number should at least divide 1 given number.
Step 2: Write the quotients right below the numbers in the next row.
Step 3: Now, for the next division step consider the in a higher place quotients equally the new dividends.
Pace 4: Recollect of a prime number again which exactly divides at least ane of the dividends.
Step 5: Repeat the steps till we get one in the final row.
Step six: Multiply all the prime number numbers on the left-hand side of the bar. That will exist the required LCM.

Permit united states of america accept an example of four numbers: 7, 8, 14, and 21, and follow the steps written below:

Footstep 1: Split up the numbers seven, viii, 14, 21 by the smallest prime number, i.due east., 2.
Stride two: Write the quotients of the divisible numbers (8 and 14) below the numbers in the next row in this mode: 7, 4, 7, and 21. Note, that for the remaining numbers 7 and 21, which are not divisible by 2, nosotros re-create the numbers as it is.
Step iii: Now for the side by side division pace, 7, 4, 7, 21 will exist the new dividends.
Stride iv: Think of the smallest prime number number again which exactly divides at least 1 of the numbers 7, 4, 7, 21.
Pace 5: Echo the process and write the caliber beneath the numbers. Hither, on dividing seven, 4, vii, 21 past 2, we go the quotients every bit seven, 2, vii, 21. [Only 4 was divisible by 2 in this pace, so nosotros copy the other three numbers every bit it is in the next row]
Pace half-dozen: Now, 7, 2, 7, 21 are the next dividends.
Step vii: Repeat the steps till we get 1.
Step viii: Multiply all the prime number numbers at the left-hand side of the bar to get the LCM of the given numbers.

Note: Dissever the numbers only by prime numbers.

LCM and HCF of numbers by division method
Therefore, the LCM of 7, 8, fourteen, and 21 is 168.

Do you lot know that for any ii numbers, if we know any one of the values of HCF or LCM, we can easily find the other without using any of the above 2 methods? LCM and HCF of two numbers share a relationship with them and with each other that we are going to learn at present.

HCF and LCM Formula

The LCM and HCF formula of two numbers 'a' and 'b' is given equally HCF × LCM = a × b. In other words, the formula of HCF and LCM states that the production of any two numbers is equal to the product of their HCF and LCM. To know more well-nigh LCM and HCF human relationship, visit this commodity.

HCF and LCM Tricks:

If 1 is the HCF of 2 numbers, and then their LCM volition be their product.
For two coprime numbers, the HCF is ever ane.

Difference between HCF and LCM

The departure between the concept of HCF and LCM will exist cleared to you lot through the following table:

HCF	LCM
Information technology stands for highest common gene.	It stands for least common multiple.
HCF is the largest of all the common factors of the given numbers.	LCM is the smallest of all the common multiples of the given numbers.
HCF of given numbers cannot be greater than whatever of them.	LCM of given numbers cannot be smaller than any of them.

Related Manufactures

Check out the interesting topics mentioned beneath to learn more than virtually HCF and LCM.

Backdrop of HCF And LCM
Least Common Multiple Formula
Factors and Multiples

Let us have a look at some solved examples on HCF and LCM.

HCF and LCM Word Problems

Example one: Find the HCF and LCM of 180 and 24 using prime factorization.

Solution:

HCF of 180 and 24:
The prime factors of 180 = 2 × ii × 3× three× 5
The prime number factors of 24 = 2 × two × 2 × three
The HCF is the product of the common prime factors of the given numbers.
Therefore, the HCF of 180 and 24 is 2 × 2 × three = 12.

LCM of 180 and 24:

The prime number factors of 180 = ii × 2 × 3 × iii× 5
The prime number factors of 24 = 2 × 2 × two × 3
Taking all the prime number factors from both the numbers only one time, nosotros have: Common prime factors (2 × 2 × 3) × Uncommon prime number factors (2 × iii× 5) = 360
Therefore, the LCM of 180 and 24 = 360.
Example 2. Find the HCF and LCM of 126 and 162 using the partitioning method.

Solution: Outset, we will notice the HCF of the two numbers 126 and 162 using the given steps:

Divide 162 by 126. The obtained balance is 36.

Make 36 as the divisor and 126 as the dividend and perform the division over again. Here the obtained remainder is xviii.

Make 18 as the divisor and 36 as the dividend and perform the division again. Here the obtained residue is 0.

The final divisor,18, is the HCF of 126 and 162.

Therefore, the HCF of 126 and 162 = 18.

Let us find the LCM of 126 and 162 by partitioning method using the following steps:
- Stride 1: Dissever the numbers 126 and 162 by the smallest prime number, i.eastward., ii.
- Step 2: Write the quotient beneath the numbers in the next row: 63 and 81.
- Step 3: Now for the next partition step, 63 and 81 will be the dividends.
- Step iv: Think of a prime once more which divides at to the lowest degree one of the two numbers 63 and 81.
- Step 5: Write the quotient below the numbers 63 and 81. The next set of quotients are 21 and 27.
- Step 6: Now 21 and 27 are the new dividends.
- Step seven: Repeat the steps till we get 1 in the final row.
- Step 8: Multiply all the prime number numbers at the left-hand side of the bar and get the LCM of the given numbers.
Therefore, the LCM of 126 and 162 is 1134.
Example three: Find the product of two numbers whose LCM and HCF are 12 and 2 respectively.

Solution: Nosotros know that the HCF and LCM formula for two numbers is HCF × LCM = product of the numbers.
Therefore, the product of numbers = 12 × 2 = 24.

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HCF and LCM Questions

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FAQs on HCF and LCM

What is the Full Form of HCF and LCM?

The total form of HCF is "Highest Mutual Factor" and the full grade of LCM is "Least Common Multiple" or "Lowest Common Multiple".

What is the Difference Between HCF and LCM?

The least mutual multiple (LCM) of ii or more numbers is the smallest number among all the mutual multiples of the given numbers, whereas, the HCF (Highest Mutual Factor) of two or more numbers is the highest number amidst all the common factors of the given numbers.

What is the Relationship Between HCF and LCM of Two Numbers?

The relationship betwixt the HCF and LCM of two numbers is that the production of the LCM and HCF of any two given numbers is equal to the product of the given numbers. Allow u.s.a. assume 'a' and 'b' are the ii given numbers. The formula that shows the relationship between their LCM and HCF is: LCM (a,b) × HCF (a,b) = a × b. For example, let us have two numbers 12 and 8. Permit us use the formula: LCM (12,8) × HCF (12,8) = 12 × 8. The LCM of 12 and 8 is 24; and the HCF of 12 and 8 is iv. Putting the values in the formula we have 24 × 4 = 12 × 8. This shows: 96 = 96.

What is the HCF and LCM of numbers?

The highest common factor (HCF) of the given numbers is the largest number which divides each of the given numbers without leaving whatsoever residuum. The least common multiple (LCM) of two or more numbers is the smallest of the common multiples of those numbers.

What is the Employ of HCF and LCM?

HCF can be used in the post-obit situations:

When we want to divide the things into smaller sections.
To arrange things in groups and rows.

LCM can be used in the following situations:

An event that is repeating continuously.
For the assay of a situation that will occur once again at the same fourth dimension.

How do you lot find the HCF and LCM in Math?

There are diverse methods to observe the HCF and LCM of numbers. The two common ways to find the LCM and HCF of the given numbers are the prime factorization method and the sectionalization method.

What are the Steps to exist Followed to Calculate the HCF and LCM of Two Numbers Using the Partitioning Method?

To find the HCF of the given numbers past division method, we follow the given steps:

Separate the given numbers (larger number by the smaller number) and check the remainder.
Make the remainder of the above step as the divisor; and the divisor of the above stride equally the dividend and perform the partitioning again.
Continue the division process till we get the rest as 0.
The terminal divisor volition be the HCF of the two numbers.

To detect the LCM of the given numbers by division method we follow the given steps:

Step ane: Carve up the numbers by the smallest prime number number.
Step two: Write the quotients correct beneath the numbers in the next row.
Step 3: Now, for the adjacent division step consider the above quotients as the new dividends.
Step 4: Think of a prime number once more which exactly divides at to the lowest degree ane of the dividends.
Step 5: Repeat the steps till we go 1 in the final row.
Stride half-dozen: Multiply all the prime numbers at the left hand side of the bar to get the LCM of the given numbers.

How to Notice HCF and LCM using Prime number Factorization?

Firstly, we detect the prime number factorization of the numbers. And then, the HCF of the given numbers will be the product of the common prime factors that occur in the prime factorization of both the numbers. And the LCM of those numbers will be the product of the common factors (taken simply once) and the uncommon or the remaining factors.

Source: https://www.cuemath.com/numbers/hcf-and-lcm/

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