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Real Numbers

Highest Common Factor

What is Highest Common Factor?

Can you guess how many things you have common with your friends? Let’s say you have 5 apples and 10 mangoes, and your friend Krishna has 3 apples and 15 mangoes – what is the highest common factor between you and Krishna? Let’s explore some of the interesting methods to find out.

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As we know, factors of a number are all the numbers that divide into that number, leaving a remainder of zero. So, for example, the factors of the number 12 are 1,2,3,4,6 and 12. So the Highest Common Factor (HCF) is nothing but the largest number that divides into two or more numbers. So when we list out all the factors of two or more numbers, the greatest number among its common factors is the HCF. It is alternatively also known as the Greatest Common Divisor (GCD).

Example:

Factors of the number 10 are 1,2,5 and 10

Factors of the number 15 are 1,3,5 and 15

HCF (10,15) = 5

Methods of Calculating Highest Common Factor

Listing all factors

The simplest method of deriving the HCF is to simply list all the factors of the given number, identify all common factors, and choose the greatest of all the common factors, arriving at your Highest Common Factor.

Example:

16 = 1, 2, 4, 8, 16

40 = 1, 2, 4, 5, 8, 10, 20, 40

HCF(16,40) = 8

Download NCERT Solutions for Class 10 Maths

Prime factorization

Here we the numbers are written as a product of prime factors. So we list prime factors of all the given numbers and find the common prime factors among them. Multiplying all the common prime factors will give you the Highest Common Factor.

Example:

20 = 2*2*5

12 = 2*2*3

HCF = 2*2 = 4

Continued Division

This method is used when we have to find the HCF of relatively larger numbers.

Highest common factor

Step 1 Divide the larger number by the smaller number
Step 2 The remainder from Step 1 becomes the divisor, and divisor of Step 1 becomes the dividend.
Step 3 Continue this division process till remainder becomes zero.
Step 4 The divisor of the last division (when remainder is zero) is your HCF.

Finding Highest Common Factors of Decimals

If the numbers given are decimal numbers, the method of finding HCF is very much similar to Prime Factorization method, with a little modification.

Example:

Say we have to find the HCF of 1.20 and 22.5

First, we convert both numbers to like decimals i.e 1.20 and 22.50

Now we express both numbers, without decimals, as a product of their prime numbers

120 = 2*2*2*3*5

2250 = 2*3*3*5*5*5

So, HCF (120, 2250) = 2*3*5 = 30

Therefore HCF (1.20, 22.5) = 0.30 (taking 2 decimal points)

Finding Highest Common Factor of Fractions

So when finding the HCF of a fraction there is an easy formula to follow

HCF= HCF of numerators/ LCM of denominators

Example:

Say we are to find the HCF of (4/5) and (3/7)

Here the HCF of the numerators 4 and 3 is 1 and the LCM of denominators 5 and 7 is 35.

Therefore the HCF of the two fractions id 1/35.

More Solved Examples for You

Question 1: Find the greatest number that divides 167 and 95, leaving 5 as remainder.

Answer : Although the question sounds complicated, the solution is a simple application of HCF.

The said number divides 167 and leaves 5 as a remainder, so the number divides 162 (167-5) perfectly.

The said number also divides 95 and leaves a remainder of 5, so the number divides 90 (95-5) perfectly.

Therefore the solution is the HCF of 162 and 90, which is 18.

Question 2: How can one find out the highest common factor?

Answer: One can find out the highest common factor by doing multiplication of all the factors whose appearance takes place in both lists. As such the HCF of 60 and 72 is 2 × 2 × 3 which turns out to be12.

Question 3: What is meant by HCF in maths?

Answer: Highest Common Factor (HCF) refers to the greatest or largest factor that is common to any two or more given natural numbers. HCF can also be called as GCD (Greatest Common Divisor).

Question 4: Find out the HCF of 50 and 70?

Answer: Factors of 70 (Seventy) happen to be 1, 2, 5, 14, 7, 10, 35 and 70. Therefore, common factors of 50 and 70 consist of 1, 2, 5, and 10. Highest common factor (H.C.F) of 50 and 70 turns out to be 10.

Question 5: What will be the HCF of 5 and 7?

Answer: The factors of 7 happen to be 7 and 1. Furthermore, the factors of 5 are 5 and 1. The common factors of 5 and 7 happen to be only 1. Therefore, the greatest common factor of the numbers 5 and 7 is 1.

 

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