2. Highest Common Factor (HCF)
2. Highest Common Factor (HCF)
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Grasp the essence of the concept of HCF, recognizing it as the largest common factor/divisor shared by two or more numbers.
Apply efficient methods, such as prime factorization and factor tree, to find the HCF of given numbers.
Find the HCF of two or more numbers using different strategies.
a. Finding Factors:
To find the factors of a given number, divide the number by each number between 1 and the number itself. The numbers that give us a remainder of 0 are factors of the number.
There are some methods of finding the factors of a number
1. Division Method:
Example 1: Find the factors of 10.
The factors of 10 are 1, 2, 5, and 10.
Example 2: Find the factors of 12.
2. Rainbow Method (Listing Method): The listing method involves the process of listing the factors of the given numbers.5
STEP 1: Write the number 1 on the left side of your work space, and the number you’re factoring—in this case, 48—on the right side. Connect them with an arc.
STEP 2: Since 48 is even, consider the next number 2. Write 2 on the left side of your work space and its factor pair—24—on the right.
STEP 3: Since 48 is divisible by 3. consider the next number, 3. Write 3 on the left and its factor pair—16—on the right. Continue this process for 4 (a factor pair with 12), 5 (not a factor, so don’t write it down), and 6 (a factor pair with 8), and 7 (not a factor). Once your factors have met in the middle of your rainbow, you can be certain you’ve found all the factors of your number.
48 has ten factors and they are 1, 2, 3, 4, 6, 8, 12, 16, 24, 48.
Example 1: Find the factors of 18.
Example 2: Find the factors of 42.
b. Common factors:
Common factors of two numbers are the factors common to (or shared by) both the numbers.
Example 1:
What are the common factors of 4 and 6?
Factors of 4: 1, 2, 4
Factors of 6: 1, 2, 3, 6
Common factors of 4 and 6: 1, 2
Example 2:
Find the common factors of 8 and 12.
Factors of 8: 1, 2, 4, 8
Factors of 12: 1, 2, 3, 4, 6, 12
Common factors of 8 and 12: 1, 2, 4
c. What is the Highest Common Factor?
c. What is the Highest Common Factor?
The highest number among the common factors identified between the two numbers is called the Highest Common factor (HCF). It is also called the Greatest Common Factor (GCF) or the Greatest Common Divisor (GCD).
Example 1: What is the HCF of 4 and 6?
Factors of 4: 1, 2, 4
Factors of 6: 1, 2, 3, 6
Common factors of 4 and 6: 1, 2
Highest common factor: 2
Example 2: Find the GCF of 8 and 12.
Factors of 8: 1, 2, 4, 8
Factors of 12: 1, 2, 3, 4, 6, 12
Common factors of 8 and 12: 1, 2, 4
Highest common factor: 4
Watch a video to understand HCF
Vid 2.1A Highest Common Factor (HCF)
Watch a video to understand HCF
Vid 2.1B Highest Common Factor (HCF)
1. Factor Tree Method
1. Factor Tree Method
A factor tree is a diagram used to break down a number by dividing it by its factors until all the numbers left are prime.2 3
Example 1: Find the factors of 120 using a factor tree.
Example 2: Find the HCF using factor trees for 18 and 27.
2. Prime Factorization Method
2. Prime Factorization Method
Prime factorization is defined as a way of finding the prime factors of a number, such that the original number is evenly divisible by these factors.
Follow the below-given steps to find the HCF of numbers using the prime factorisation method. Let us understand that using an example.
Example 1: Find the HCF of 24 and 36.
Example 2: Find the HCF of 27, 36 and 80 using prime factorization method.
Factors of 27 = 3 x 3 x 3
Factors of 36 = 2 x 2 x 3 x 3
Factors of 80 = 2 x 2 x 2 x 2 x 5
As there are no common factors in all 3 numbers, the HCF is 1.
Example 3: Find the HCF of 126, 162 and 180 using prime factorization method.
Factors of 126 = 2 x 3 x 3 x 7
Factors of 162 = 2 x 3 x 3 x 3 x 3
Factors of 180 = 2 x 2 x 3 x 3 x 5
The common factors are 2, 3 and 3
∴ H.C.F = 18
Watch a video to understand this better
Vid 2.2 HCF by Prime Factorisation
Summary :-
1. Common factors of two numbers are the factors common to (or shared by) both the numbers.
2. The highest number among the common factors identified between the two numbers is called the Highest Common factor (HCF). It is also called the Greatest Common Factor (GCF) or the Greatest Common Divisor (GCD).
3. HCF can be found using the following methods:
Factor Tree Method: A factor tree is a diagram used to break down a number by dividing it by its factors until all the numbers left are prime.
Prime Factorization Method: Prime factorization is defined as a way of finding the prime factors of a number, such that the original number is evenly divisible by these
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