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HCF of Two Numbers is the highest possible number that divides both numbers completely.HCF is the abbreviation for Highest Common Factor. The Highest Common Factor (HCF) is also referred to as the Greatest Common Divisor (GCD) or Greatest Common Factor (GCF). The HCF of two natural numbers x and y is the largest possible number that divides both x and y. For instance, the HCF of 18 and 27 is 9, as 9 is the highest factor that is common to both 18 and 27. HCF of two numbers can be calculated by the division method or prime factorization.
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Key Terms: HCF of Two Numbers, HCF, Highest Common Factor, Prime Factorization, Division, Factors, Prime Factors
What is HCF?
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HCF (Highest Common Factor) of two or more numbers refers to the highest number among all the common factors of the given numbers. HCF of two numbers is the highest factor that can divide the two numbers evenly. HCF can be calculated for two or more than two numbers. It refers to the greatest divisor for any two or more numbers, that can equally or completely divide the given numbers.
Example: The HCF of 60 and 75 is 15.
- Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60.
- Factors of 75: 1, 3, 5, 15, 25, and 75.
Here, 15 is the largest number that can divide both 60 and 75 exactly. Thus, the HCF of 60 and 75 is 15.
Check More:
| Relevant Concepts | ||
|---|---|---|
| Relation Between HCF and LCM | Perfect Numbers | Quotient |
| Properties of LCM and HCF | Multiplication and Division of Integers | Euclid's Division Lemma |
HCF of Two Numbers
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HCF of Two Numbers is the greatest factor that divides both the numbers completely without leaving any remainder. HCF of two numbers can be calculated through various methods. However, irrespective of the method, the answer to the HCF of two numbers would always be the same.
HCF of two numbers can be calculated through two main methods:
- HCF by Prime Factorization
- HCF by Division Method
Highest Common Factor (HCF)
Read More: LCM of Two Numbers
HCF of Two Numbers by Prime Factorization
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Consider that two numbers a and b are exactly divided by the largest possible number, thus, that number is said to be the highest common factor of the given numbers. The product of the common prime numbers represents the prime factorization of those numbers. To find the HCF of two numbers by the prime factorization, the following steps need to be followed:
- Step 1: Find out the prime factorization of given numbers individually.
- Step 2: Take out the common prime factors of the two numbers.
- Step 3: The product of common prime factors is the HCF of the given two numbers.
Solved ExampleExample: Find the HCF of 56 and 84 using the prime factorization. Solution: First, we will find out the prime factorization of 56 and 84 separately.
HCF of 56 and 84 will be the product of common prime factors with the lowest exponential power. HCF of 56 and 84 = 7 × 2 × 2 = 28 Thus, the HCF of 56 and 84 is 28. |
Read More: NCERT Solutions for Class 6 Maths Playing With Numbers
HCF of Two Numbers by Division Method
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In order to find the HCF of two numbers by division method, follow the given steps:
- First, divide the larger number by the smaller number.
- Now, make the remainder of the above step the divisor and the divisor of the above step as the dividend and do the long division.
- Continue with the long division in the same manner till the remainder becomes 0.
- The last divisor that is left when the remainder is zero is the HCF.
Solved ExampleExample: Calculate the HCF of 120 and 100 using long division. Solution: The two numbers are given as 120 and 100. We will first divide 120 by 100. 120/100 → 1; Remainder is 20. Dividing the first divisor 100 by the first remainder 20. 100/20 → 5; Remainder is 0. Thus, the HCF of 120 and 100 is 20. |
HCF of 120 and 100
HCF of Three Numbers
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HCF of Three Numbers can also be calculated in a similar manner as the HCF of two numbers. The HCF of three numbers can be calculated through either long division or prime factorization. Both methods would give the same value of HCF.
Solved ExampleExample: What will be the HCF of 126, 162, and 180? Solution: Through prime factorization, the factors of 126, 162, and 180 are:
Taking out the common factors of 126, 162, and 180, we get HCF (126, 162, 180) = 2 x 3 x 3 = 18 |
Also Check:
| Related Topics | ||
|---|---|---|
| LCM Formula | Divisibility Rules | Fundamental Theorem Of Arithmetic |
| Synthetic Division | Remainder Theorem | Prime Number Formula |
Things to Remember
- HCF of two numbers is the largest common number that divides both numbers without leaving any remainder.
- HCF of two numbers is always the factor of each of the numbers.
- HCF of two numbers is less than or equal to each of the numbers.
- The HCF of two prime numbers is always 1 as they do not have any other common factor.
- Division method and prime factorization are the two methods that are used to calculate the HCF of two numbers.
Sample Questions
Ques. Find the HCF of 24 and 36 using prime factorization. (2 Marks)
Ans. The prime factorization of 24 and 36 is as follows:
- 24 = 2 x 2 x 2 x 3
- 36 = 2 x 2 x 3 x 3
The product of the common factors is 2 x 2 x 3 = 12.
Thus, the HCF (24, 36) is 12.
Ques. Find the HCF of 35 and 55 using prime factorization. (2 Marks)
Ans. We will use prime factorization to write the factors of 35 and 55,
- 35 = 5 x 7
- 55 = 5 x 11
It can be seen that the highest common factor for 35 and 55 is 5.
Thus, the HCF (35, 55) is 5.
Ques. What will be the HCF of 45 and 60 by the division method? (2 Marks)
Ans. First, we will divide 60 by 45.
60/45 → 1; Remainder is 15.
Now, dividing 45 by 15, we get
45/15 → 3; Remainder is 0.
Thus, the HCF of 45 and 60 is 15
Ques. Using prime factorization, find the HCF of 72 and 120. (2 Marks)
Ans. In the prime factorization method, we will first write down all the prime factors of given numbers and then find the highest common factor among the two.
- 72 = 2 × 2 × 2 × 3 × 3 = 23 × 32
- 120 = 2 × 2 × 2 × 3 × 5 = 23 × 3 × 5
HCF of 72 and 120 will be the product of common prime factors with the lowest exponential power, thus, 23 x 3. Thus, HCF 72 and 120 = 2 x 2 x 2 x 3 = 24
Ques. What will be the HCF of 399 and 437 by division method? (2 Marks)
Ans. First, we will divide the larger number 437 by the smaller number 399 and get the remainder. Now, the remainder of the above step is made the divisor, and the divisor of the above step is made the dividend. Continue the division similarly till the remainder becomes 0.
The last divisor 19 will have zero as the remainder and will be the HCF of two numbers 399 and 437.
Ques. Find the HCF of two numbers 150 and 225. (2 Marks)
Ans. The largest number that divides any two given numbers exactly without any remainder is called the HCF (Highest Common Factor) of that two numbers. Therefore, the HCF of 150 and 225 is 75 as it is the largest common factor of both.
Ques. Find the HCF of two numbers 135 and 225 by Prime Factorization. (2 Marks)
Ans. The prime factorization of 135 and 225 is
- 135 = 3 × 3 × 3 × 5
- 225 = 3 × 3 × 5 × 5
Thus, the HCF of 135 and 225 = 3 x 3 x 5 = 45
Ques. What will be the HCF of two numbers 2 and 8? (2 Marks)
Ans. The first step would be to find out the prime factors of 2 and 8. The prime factorization of 2 is 2 and the prime factorization of 8 is 2 × 2 × 2, thus, the common prime factor is 2. Therefore, the HCF of 2 and 8 is 2.
Ques. What will be the HCF of 0 and 6? (2 Marks)
Ans. The HCF of 0 and 6 is undefined. We know that the factors of 6 are 1, 2, 3, and 6, however, there are no factors of 0. Thus, we can not find out the common factors of 0 and 6, therefore the HCF of 0 and 6 is not defined.
Ques. What is the other name for H.C.F? (1 Mark)
Ans. HCF is also referred to as the Greatest Common Factor (GCF) and Greatest Common Divisor (GCD).
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