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HCF of Two Numbers

The Highest Common Factor(HCF) of two numbers is the highest possible number which divides both the numbers exactly. The highest common factor (HCF) is also called the greatest common divisor (GCD). Let's discuss more about HCF of two numbers, how to find the HCF of two numbers by prime factorization and division method. Let us explore the methods to find the HCF of two numbers.

1. HCF of Two Numbers by Prime Factorization
2. HCF of Two Numbers by Division Method
3. Solved Examples on HCF of Two Numbers
4. Practice Question on HCF of Two Numbers
5. FAQs on HCF of Two Numbers

HCF of Two Numbers by Prime Factorization

When two numbers, say p and q are exactly divided by the largest possible number, the number is said to be the highest common factor of the respective numbers. When we show the product of two prime numbers, let's say x and y, that is their prime factorization. This means the product of two prime numbers represents prime factorization. Let's understand, how to find the HCF of two numbers by the prime factorization method. Following are the steps we need to follow:

  • Find the prime factorization of given numbers individually.
  • List out the common prime factors of those numbers.
  • Product of common prime factors is the HCF of given numbers.

For better understanding, let's solve some examples of HCF of two numbers by prime factorization. We will find the HCF of 56 and 84. Let's represent the numbers using the prime factorization.
So, we have, 56 = 2 × 2 × 2 × 7 and 84 = 2 × 2 × 3 × 7 . Now, HCF of 56 and 84 will the product of common prime factors with the lowest exponential power, that is, 7 and 2². So, HCF of 56 and 84 = 7 × 2 × 2 = 28.

HCF of Two Numbers by Division Method

To find HCF of two numbers by division method, we need to follow the following steps.

  • Divide the larger number by the smaller number. 
  • Make the remainder of the above step as the divisor and the divisor of the above step as the dividend and do the long division again.
  • Continue the long division till the remainder becomes 0.
  • HCF is the last divisor left, when remainder = 0 

For example, let's find HCF of 56 and 84 by the division method.

HCF of two numbers 56 and 84 by division method

Important points

  • HCF of two numbers divides each of the numbers without leaving any remainder.
  • HCF of two numbers is a factor of each of the numbers.
  • HCF of two numbers is always less than or equal to each of the numbers.
  • HCF of two prime numbers is 1 always.
  • HCF of two numbers can also be found out with help of the listing factors method. In this method, we list out all the factors of two numbers and take out the highest common factor. 

Related Articles on HCF of Two Numbers

Important Topics
 
LCM - Least Common Multiple
Factors
 

Solved Examples

  1. Example 1: Find the HCF of 72 and 120.

    Solution:
    The given numbers are 72 and 120. To find the HCF of two numbers 72 and 120 we can use the prime factorization method. In the prime factorization method firstly we need to write all the prime factors of respective numbers and then find the highest common factor among the two. Here we have,

    • Prime factorization of 72 = 2 × 2 × 2 × 3 × 3 or 23 × 32
    • Prime factorization of 120 = 2 × 2 × 2 × 3 × 5 or 23 × 3 × 5

     Therefore, HCF of 72 and 120 will the product of common prime factors with the lowest exponential power, that is, 23 x3 so, HCF 72 and 120= 2 x 2 x 2 x 3 = 24

  2.  

    Example 2: Find the HCF of 399 and 437 by division method.

    Solution: 

    To find HCF by division method, we need to follow the following steps.

    • Divide the larger number  437 by the smaller number 399 and get the remainder. 
    • Make the remainder of the above step as the divisor and the divisor of the above step as the dividend and do the long division again.
    • Continue the long division till the remainder becomes 0.
    • The last divisor 19 will be the HCF of two numbers 399 and 437.

    Finding HCF of two numbers 399 and 437 by division method

    Therefore, HCF of 399 and 437 is 19.

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FAQs on HCF of Two Numbers

What Is the HCF of Two Numbers 150 and 225?

The largest possible number which divides the given numbers exactly without any remainder is called the HCF (Highest Common Factor). Thus, HCF of 150 and 225 is 75.

What Is the HCF of Two Numbers 135 and 225 by Prime Factorization?

HCF (Highest Common Factor) of two or more numbers is the largest possible number that divides the given numbers leaving the remainder 0. HCF of two numbers 135 and 225 by prime factorization is 45.

How to Find HCF of Two Numbers by Prime Factorization?

Prime factorization of a given number is representing that number as the product of prime numbers. Let's understand, how to find the HCF of two numbers by the prime factorization method. Following are the steps we need to follow:

  • Step 1: Find the prime factorization of each number
  • Step 2: Find the common prime factors of those numbers
  • Step 3: Find the HCF of those numbers by finding the product of the prime factors with the lowest exponential power or factors that are common to each of the given numbers.

How Do You Find HCF of Two Numbers by Division Method?

To find the HCF of two numbers by division method, we need to follow the following steps.

  • Divide the larger number by the smaller number. 
  • Make the remainder of the above step as the divisor and the divisor of the above step as the dividend and do the long division again.
  • Continue the long division till the remainder becomes 0.
  • HCF is the last divisor which leaves the remainder 0. 

Find the HCF of Two Numbers 2 and 8.

First we will find the prime factors of 2 and 8. Prime factorization of 2 is 2 and prime factorization of 8 is 2 × 2 × 2. Common prime factor = 2. Therefore, HCF of 2 and 8 is 2.

What Is the HCF of 0 and 6?

HCF of 0 and 6 are undefined. The factors of 6 are coming: 1, 2, 3, and 6 There are no factors of 0. Since we can't find out the common factors of 0 and 6, therefore the HCF of 0 and 6 is not defined.

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