Answer sheets of meritorious students of class 12th’ 2012 M.P Board – All SubjectsFebruary 14, 2013
39 Insightful Publications
The largest common factor of two or more numbers is called the highest common factor or HCF. LCM and HCF are two such concepts that find importance not only for school-level Mathematics but also in various other exams, like CAT, MAT, recruitment exams for government jobs, etc.
It is, therefore, important that you understand the HCF definition and how to find the HCF of given numbers. Other names for HCF are Greatest Common Divisor (GCD) and Greatest Common Measure (GCM). We will discuss all the methods to find the HCF along with some solved examples and practice questions in this article. Read on to find out!
The HCF of a set of whole numbers is the largest positive integer that divides evenly all the given numbers with zero remainders. In other words, there is no integer bigger than the HCF which will be a common divisor for any given set of numbers.
For example, for the set of numbers 18, 30, and 42, the HCF = 6.
What is a Factor?
Factors of any number are numbers that we can multiply to get the original number:
A number can have multiple factors. For example, 12 has 1, 2, 3, 4, 6, and 12 as factors.
What are Common Factors?
A common factor for two or more numbers is any number that divides all the given numbers. For example,
Factors of 12 are 1, 2, 3, 4, 6 and 12
Factors of 30 are 1, 2, 3, 5, 6, 10, 15 and 30
In these factors, we can see that 1, 2, 3, 4, and 6 are common to both 12 and 30. So, we say that 1, 2, 3, 4, and 6 are common factors for 12 and 30.
After we get common factors for any given set of numbers, we can easily find the GCD. The common factor which is the largest is the GCD or HCF for those numbers. In this case, 6 will be the GCD.
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There are mainly four methods to find the HCF of two or more numbers.
1. Factorisation Method
2. Prime Factorisation Method
3. Division Method
4. Shortcut Method
Let us understand each method with examples.
HCF by factorisation method is one of the most common methods to find HCF of a given set of numbers. In this method, we find the greatest common factor by listing down all the factors of the numbers.
The step-by-step process on how to find the HCF by factorisation method is listed below:
1st step: List down all the factors of given numbers.
2nd step: Among all the listed factors, look for the largest common factor. That number will be the HCF of those numbers.
Let us understand this method with an example.
Example: Find the HCF of 12 and 16.
Write each number as a product of its factors. So, we get:
Factors of 12 are 1, 2, 2, and 3. Factors of 16 are 1, 2, 2, 2, and 2. The biggest common factor is 2×2. So, the HCF of 12 and 16 is 4.
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The prime factorisation method is also called the factor tree method. Let us understand how to find out HCF using this method with an example:
1st step: Write each number as a product of its prime factors.
2nd step: Now list the common factors of both the numbers.
3rd step: The product of all common prime factors is the HCF (use the lower power of each common factor).
Let’s understand this method with an example.
Example: Find the HCF of 16 and 24.
Write each number as a product of its prime factors.
16 = 24
24 = 23 x 3
We know that the product of all common prime factors is HCF. The common prime factor in this example is 23 [∵ 24 can be written as 23 x 2]
So, HCF = 23 = 8.
How to find HCF by long division method? In this method, we divide the given numbers, simultaneously, to get the common factors between them. This method is comparatively easy but lengthy. The step-by-step process on how to find HCF by division method is listed below:
1st step Write the given numbers horizontally, by separating them with commas.
2nd step: Find the smallest prime number which can divide the given numbers. The remainder should be 0 on dividing those numbers by that small number (write on the left side).
3rd step: Now write the quotients.
4th step: Repeat the process, until you reach the stage, where there is no prime number that can divide all the numbers exactly.
5th step: Write down all the common prime factors on the left side. The product of these common prime factors is the HCF of the given numbers.
Let us understand how to find the HCF by division method using an example.
Example: Find the highest common factor of 18 and 24.
We can see that the prime factors on the left side divide all the numbers exactly. So, they all are common prime factors. We have no common prime factor for the numbers at the bottom.
So, HCF = 2 × 3 = 6
There is a shortcut method to find the HCF of numbers quickly. The step-by-step process on how to find HCF quickly is explained below:
We have explained how to find the HCF of three numbers by using the long division method. The step-by-step process is listed below:
1st step: Calculate the HCF of the first 2 numbers.
2nd step: Find the HCF of the 3rd number and the HCF found in Step 1.
3rd step: The HCF you got in Step 2 will be the HCF of the given 3 numbers.
You can also find the HCF of more than 3 numbers using the method explained above.
Here we have provided some of the solved examples on the HCF.
|Question 1: Find the HCF of 15 and 24.|
Solution: Let us first find the prime factors of 15 and 24:
|15 = 3 X 5|
24 = 2 X 2 X 2 X 3
As 3 is the only factor that is common for both 15 and 24, HCF = 3.
|Question 2: Find the HCF of 4/9 and 6/21.|
Solution: Numerators of the two fractions: 4 and 6
Prime factors of 4 and 6:
|4 = 2 X 2|
6 = 2 X 3
HCF of 4 and 6 is 2.
Denominators of the two fractions: 9 and 21
Prime factors of 9 and 21:
|9 = 3 X 3|
21 = 3 X 7
HCF of 9 and 21 is 3. LCM of 9 and 21: 3 X 3 X 7 = 63.
Hence, HCF of 4/9 and 6/21 = HCF of Numerators/LCM of Denominators
|Question 3: Two numbers are in the ratio of 5:11. If their HCF is 7, find the numbers.|
Solution: Let the numbers be 5x and 11x. Since 5:11 is already the reduced ratio, ‘x’ has to be the HCF. So, the numbers are 5 x 7 = 35 and 11 x 7 = 77.
|Question 4: Find the greatest number which on dividing 70 and 50 leaves remainders 1 and 4 respectively.|
Solution: The required number leaves remainders 1 and 4 on dividing 70 and 50 respectively. This means that the number exactly divides 69 and 46.
So, we need to find the HCF of 69 (3 x 23) and 46 (2 x 23).
HCF (69, 46) = 23
Thus, 23 is the required number.
|Question 5: A rectangular field of dimension 180m x 105m is to be paved by identical square tiles. Find the size of each tile and the number of tiles required.|
Solution: We need to find the size of a square tile such that a number of tiles cover the field exactly, leaving no area unpaved.
For this, we find the HCF of the length and breadth of the field.
HCF (180, 105) = 15
Therefore, size of each tile = 15m x 15m
Also, the number of tiles = area of field/area of each tile
=> Number of tiles = (180 x 105) / (15 x 15)
=> Number of tiles = 84
Hence, we need 84 tiles, each of size 15m x 15m.
Here we have provided some of the practice questions related to the HCF:
|Q1: What is the greatest number which divides 639, 1065 and 1491 exactly? |
Q2: What is the H.C.F. of 4/9, 10/21 and 20/63?
Q3: The H.C.F. of two numbers is 12 and their difference is 12. What are the numbers?
Q4: The HCF of two numbers is 29 & their sum is 174. What are the possible numbers?
Q5: Find the side of the largest square slab which can be paved on the floor of a room 5 meters 44cm long and 3 meters 74 cm broad.
Q6: The product of two numbers is 6760 and their H.C.F. is 13. How many such pairs can be formed?
Q7: 3 different pieces of iron are of varying length are given to a student which are 44cm, 22 cm,55 cm respectively. He has to form rods of maximum length such that no iron waste is left. Find the maximum length of such a rod.
Some of the important properties of HCF are as under:
a. The HCF of given numbers is never greater or more than any of the numbers.
b. The HCF of two or more prime numbers is always 1.
c. The product of two numbers, a and b, is equal to the product of their HCF and LCM. This means:
|a x b = LCM of (a & b) x HCF of (a & b)|
d. HCF of Fractions = HCF of Numerators/LCM of Denominators
HCF of a set of whole numbers is the biggest positive integer that divides all the given numbers with zero remainders. HCF of numbers can be calculated using four methods. The four methods to calculate the HCF of numbers are factorisation, prime factorisation, division, and HCF by shortcut method. Furthermore, it is important to note that the HCF of prime numbers is 1. The HCF of any number or set of numbers comprising zero is undefined.
Now you are provided with all the necessary information regarding the greatest common factors. Practice more questions and master this concept.
Students can make use of NCERT Solutions for Maths provided by Embibe for their exam preparation.
Free Practice Questions and Mock Tests for Maths (Class 8 to 12)
Following are the frequently asked questions on HCF:
Q1: How to calculate HCF?
A: To find the HCF of two numbers, we can use prime factorisation or the long division method. The methods are explained in detail in this article.
Q2: Are HCF, GCD, and GCF the same?
A: Yes, all the terms refer to the same thing. The highest common factor means the largest number which can exactly divide a set of numbers.
Q3: What is HCF meaning?
A: The largest positive integer that divides evenly a set of numbers with zero remainders is the HCF of those numbers. For example, 8 and 12 have common factors of 1, 2, and 4. The highest common factor is 4.
Q4: Can 1 be an HCF?
A: Yes. Co-prime numbers do not have any common factor between them. So, the HCF of those numbers is 1.
Q5: What is the HCF of 0 and 12?
A: The HCF of any set of numbers containing 0 is ‘undefined’.
Q6: What is the HCF of three numbers?
A: HCF of 3 numbers = HCF of [HCF of first 2 numbers and 3rd number]
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