How To Find Lcm And Hcf By Prime Factorisation Method
The two principal methods which are used to discover the LCM (Least Mutual Multiple) and the HCF (Highest Common Factor) of the numbers are the Prime Factorization Method and Partitioning Method. Both the methods are explained here with many examples. We have provided the prime number factors of the given numbers, such as 24, 12, thirty, 100, etc. using these methods. Here, you will acquire how to find the LCM and HCF of the numbers by both the approaches. Before that, let united states of america talk over what are LCM and HCF in detail.
Least Mutual Multiple (LCM)
The least or smallest mutual multiple of whatsoever two or more given natural numbers are termed every bit LCM.
For instance, LCM of 10, fifteen and 20 is lx.
Highest Common Gene (HCF)
The largest or greatest gene common to whatever two or more given natural numbers is termed every bit HCF of given numbers. It is also known as GCD (Greatest Common Divisor).
For example, HCF of iv, 6 and viii is 2.
How to Find LCM and HCF?
We tin can detect HCF and LCM of given natural numbers by two methods i.e., by prime factorization method or segmentation method. In the prime number factorization method, given numbers are written as the product of prime factors. While in the division method, given numbers are divided by the to the lowest degree mutual gene and continue still rest is nil.
Note: Prime numbers are numbers which have but 2 factors i.e. ane and the number itself.
LCM past Prime number Factorization Method
Here, given natural numbers are written as the product of prime factors. The lowest common multiple will be the product of all prime factors with the highest caste (power).
Example one:
Find the LCM of 20 and 12 by prime factorization method.
Solution:
Step 1: To find LCM of twenty and 12, write each number as a product of prime number factors.
\(\brainstorm{assortment}{l}xx=ii\times 2\times five=ii^{two}\times 5 \\ 12=2\times 2\times 3=2^{2}\times 3\end{assortment} \)
Step 2: Multiply all the prime factors with the highest degree.
Hither we have ii with highest power 2 and other prime factors 3 and v. Multiply all these to get LCM.
LCM of xx and 12 =
\(\begin{array}{l}ii\times 2\times 3\times v=ii^{2}\times 3\times 5=lx\stop{array} \)
LCM by Sectionalisation Method
In this method, split up the given numbers by mutual prime until the remainder is a prime or one. LCM will be the production obtained by multiplying all divisors and remaining prime numbers.
Example 2:
Find the LCM of 24 and 15 by the segmentation method.
Solution:
Pace 1: Divide the given numbers by the least prime number.
Here, 2 is the least number which will carve up 24.
Step 2: Write the quotient and the number which is not divisible by the above prime number in the second row.
In the second row, write the caliber we get after the partition of 24 by 2. Since 15 is not divisible by 2, write fifteen in the second row as it is.
Stride three: Dissever the numbers with some other least prime.
Step four: Proceed sectionalisation until the residual is a prime number or 1.
Pace 5: Multiply all the divisors and remaining prime number (if whatsoever) to obtain the LCM.
LCM of 24 and 15=
\(\begin{array}{l}2\times 2\times 2\times 3\times 5=2^{three}\times three\times five=120\cease{array} \)
What is the LCM of 60, 84, and 108?
The LCM of 60, 84, and 108 is 3780. The LCM of 60, 84, and 108 is the least common multiple of 60, 84, and 108. The LCM of 60, 84, and 108 is found using iii different methods such every bit prime number factorization method, listing the multiples, and the sectionalization method.
Now, let us hash out all these three methods one past one to discover the LCM of sixty, 84, and 108.
LCM of sixty, 84, and 108 by Listing Multiples
Step one: List the multiples of sixty, 84, and 108:
Multiples of 60: 60, 120, 180, 240, 300, 360, 420, 480, 540, ….
Multiples of 84: 84, 168, 252, 336, 420, 504, 588, 672, …
Multiples of 108: 108, 216, 324, 432, 540, 648, 756, 864, 972, ….
Step two: Find the common multiples from the multiples of threescore, 84, and 108:
The common multiples are 3780, 7560, …
Stride three: Determine the smallest mutual multiple from the multiples of 60, 84, and 108:
Thus, the least common multiple of 60, 84, and 108 is 3780.
(i.due east.) LCM of threescore, 84, and 108 is 3780.
LCM of 60, 84, and 108 by Prime Factorization Method
To find the LCM of threescore, 84, and 108 using the prime number factorization method, start write the prime factorization of 60, 84, and 108.
Therefore, the prime factorization of 60 is 2 ii × 3 1 × 5 1
The prime factorization of 84 is 2 2 × three 1 × 7 1
The prime factorization of 108 is 2 ii × iii three
Thus, the LCM of 60, 84, and 108 is obtained by multiplying the prime factors raised to their corresponding highest power.
(i.due east) The least mutual multiple of 60, 84 and 108 = two 2 × 3 3 × 5 ane × seven i = 3780.
Hence, LCM of 60, 84, and 108 is 3780.
LCM of 60, 84, and 108 past Division Method
To find the LCM of threescore, 84, and 108 using the segmentation method, divide the numbers 60, 84, and 108 by the prime factors. So, the product of the divisor gives the result of LCM of threescore, 84, and 108.
Therefore, LCM of 60, 84 and 108 = two × 2 × 3 × 3 × 3 × v × seven = 3780.
HCF By Prime number Factorization Method
Given natural numbers to be written as the product of prime factors. To obtain the highest common factor multiply all the common prime factors with the lowest caste (power).
Instance 1:
Find the HCF of 20 and 12 by the prime number factorization method.
Solution:
Stride 1: To find HCF of 20 and 12, write each number as a product of prime factors.
20 = 2 × two × 5 = 22 × 5
12 = ii × 2 × three = 22 × three
Stride 2: Multiply all the common prime number factors with the lowest degree.
Hither we have merely 2 as a mutual prime gene with the lowest power of 2.
HCF of 20 and 12 = 22 = 4
HCF by Partitioning Method
In this method divide the largest number by the smallest number amongst the given numbers until the residuum is aught. The last divisor will be the HCF of given numbers.
Instance 2:
Find the LCM of 24 and xv by the division method.
Solution:
Step 1: Divide the largest number by the smallest number.
Here, the largest number is 24 and the smaller 1 is 15. Divide 24 by fifteen
Footstep 2: Have divisor as new dividend and balance as the new divisor, i.eastward. carve up the first divisor by the commencement residuum.
Step 3: Keep till the remainder is zero and the last divisor will be the HCF of the given numbers.
Therefore, HCF of 24 and 15 is 3.
Alternatively, nosotros can divide both the numbers by the least common prime factor, still in that location is no more mutual prime factors. Multiply all divisors to get the HCF of given numbers.
Consider the above example, HCF of 24 and 15 can also exist calculated using the following steps:
Stride one: Carve up the given numbers by the least common prime gene.
Here, iii is the least common prime number factor of 24 and 15.
Step 2: Proceed all the same at that place is no more common prime cistron. Then multiply all the divisors.
Division of 24 and xv by three will exit viii and 5 equally their remainders respectively. 8 and 5 do not take a common prime number factor.
Hence, the HCF of 24 and xv is 3.
Stay tuned with BYJU'S – The Learning App to learn more than most the division method, prime factorization method, and much more.
Oftentimes Asked Questions on Prime Factorization and Division Method for LCM and HCF
What are the dissimilar methods to observe the LCM of numbers?
The dissimilar methods to find the LCM of numbers are:
Prime Factorization Method
Division Method
List the Multiples of numbers.
What is the LCM of 60, 84, and 108?
The LCM of sixty, 84, and 108 is 3780.
How to find the LCM of numbers using the prime factorization method?
Step 1: Write the prime number factorization of the numbers.
Step ii: The LCM of the given number is obtained by multiplying the prime number factors raised to their respective highest power.
How to find the HCF of numbers using the prime number factorization method?
Step one: Write the prime number factorization of the numbers.
Step two: Detect the common prime number factors.
Step 3: The HCF of the given number is the product of common prime factors with the lowest exponential power.
What is the LCM of 36 and 48?
The LCM of 36 and 48 is 144.
How To Find Lcm And Hcf By Prime Factorisation Method,
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