Highest Common Factor (HCF)
Introduction to HCF
The Highest Common Factor (HCF), or the Greatest Common Divisor (GCD), is the biggest number that can evenly divide two or more numbers without a remainder. It is an important idea in number theory and has numerous uses in reducing fractions, determining common denominators, and solving algebraic and geometric problems. In this self-study syllabus, we will discuss its definition, techniques of computing the HCF, its properties, and everyday applications in mathematics.
Table of Content:
- Objective of Learning HCF
- Definition and Understanding of HCF
- Techniques of HCF Calculation
- Properties of HCF
- Relationship Between HCF and LCM
- Applications of HCF
- Practice Problems
- Fun Facts
- Conclusion
- FAQs
Objective of Learning HCF
1. Understand the definition of HCF and its significance.
2. Learn and apply different methods for finding HCF.
3. Solve problems related to HCF.
4. Use HCF in simplifying fractions, solving algebraic problems, and other mathematical contexts.
5. Explore the relationship between HCF and LCM (Least Common Multiple).
Definition and Understanding of HCF
HCF is the highest number that can divide two or more given numbers without a remainder. Note that the HCF of two or more numbers is always a divisor of every number in the set.
Example: Determine the HCF of 12 and 18.
- Divisors of 12 are 1, 2, 3, 4, 6, 12.
- The divisors of 18 are 1, 2, 3, 6, 9, 18.
- Common divisors are 1, 2, 3, and 6.
- Thus, HCF of 12 and 18 is 6.
Techniques of HCF Calculation
There are various techniques of HCF calculation of two or more numbers. Each technique gives a convenient method to find the HCF based on the involved numbers.
Method of Prime Factorization
In the method of prime factorization, we express each number as the product of its prime factors. The HCF of the numbers is the product of the lowest powers of all the common prime factors.
Example: Determine the HCF of 24 and 36.
- Prime factorization of 24 is 2² × 3.
- Prime factorization of 36 is 2² × 3².
- The common prime factors are 2² and 3.
- Hence, the HCF of 36 and 24 is 2² × 3 = 12.
Division Method of HCF
The division method of HCF is to divide the greater number by the smaller number and then divide the divisor by the remainder. This is done until the remainder is zero. The divisor at this stage is the HCF.
Example: Calculate the HCF of 56 and 98 using the division method of HCF.
- Divide 98 by 56: 98 ÷ 56 = 1, remainder = 42.
- Now divide 56 by 42: 56 ÷ 42 = 1, remainder = 14.
- Divide 42 by 14: 42 ÷ 14 = 3, remainder = 0.
- Because the remainder now is zero, the divisor is 14, which is the HCF.
Listing Divisors Method
In the listing divisors method, we list all the divisors of both numbers and then find the largest number common to both lists. This is the HCF.
Example: Find the HCF of 15 and 25.
- The divisors of 15 are 1, 3, 5, 15.
- The divisors of 25 are 1, 5, 25.
- The common divisors are 1 and 5.
- Hence, the HCF of 15 and 25 is 5.
Properties of HCF
HCF of two or more numbers possesses certain properties that make it helpful in the solution to various mathematical problems.
Certain valuable properties of HCF are:
1. The HCF of two numbers is less than or equal to the smaller number.
2. The HCF of a number and 1 is always 1.
3. The HCF of two co-prime numbers (numbers that have only one common factor, i.e., 1) is 1.
4. The HCF of the numbers remains the same if we subtract or add the same number to both numbers.
5. The HCF of two numbers is distributive over addition and subtraction.
Relationship Between HCF and LCM
The Highest Common Factor (HCF) and Least Common Multiple (LCM) are connected with the following equation:
HCF × LCM = Product of the two given numbers.
Example: Determine the HCF and LCM of 12 and 18.
- We already know the HCF of 12 and 18 is 6.
- The product of 12 and 18 is 216.
- Thus, LCM = 216 ÷ HCF = 216 ÷ 6 = 36.
- Thus, the LCM of 12 and 18 is 36.
Applications of HCF
There are a number of applications of HCF in mathematics, science, engineering, and everyday situations.
Listing Divisors Method
HCF is applied to simplify fractions by dividing both the numerator and denominator by their HCF.
Example: Simplify the fraction 36/48.
- The HCF of 36 and 48 is 12.
- Both 36 and 48 are divided by 12: 36 ÷ 12 = 3, 48 ÷ 12 = 4.
- Hence, 36/48 becomes 3/4.
Word Problems
HCF comes in handy when solving word problems that include dividing objects into equal portions or spotting patterns in arrangements.
Example: A carpenter has 48 inches of wood and needs to divide it into equal smaller pieces. What is the maximum possible length of each piece if there should be no remainder?
- The solution is the HCF of 48 and the number of pieces.
Practice Problems
Practice the following problems to reinforce the concepts of HCF:
Having solidified the concepts of HCF, practice the following problems:
1. Determine the HCF of 30 and 45.
2. Determine the HCF of 50, 60, and 75.
3. Determine the HCF of 72 and 108 by using the division method of HCF.
4. Reduce the fraction 120/150.
5. Solve a word problem on HCF.
Fun Facts
1. HCF vs. LCM: Whereas HCF is finding the greatest common divisor of two numbers, LCM (Least Common Multiple) is finding the smallest multiple that two numbers have in common. They are usually paired to work problems dealing with fractions or ratios.
2. HCF is Like "Dividing the Cake": Imagine the HCF to be the largest piece of cake you can cut from two cakes without cutting any pieces waste. Both cakes (numbers) must be divided into equal pieces, and the HCF provides you with the largest piece!
3. The "1" Rule: Two numbers are co-prime if they do not have any factors in common except 1. The HCF of co-prime numbers is always 1. Thus, if two numbers have an HCF of 1, it implies they are co-prime.
4. HCF and Simplification: HCF plays a major role when reducing fractions. To reduce a fraction, divide the numerator and the denominator by their HCF. This guarantees that the fraction is reduced to its simplest form.
5. Ancient Calculation Method: The process of calculating HCF, particularly the Euclidean Algorithm, was first outlined by ancient Greek mathematician Euclid more than 2,000 years ago!
Conclusion
The Highest Common Factor (HCF) is a basic idea that provides the basis for much advanced mathematics. Learning the various ways of computing the HCF and working with its characteristics, students can solve an enormous variety of mathematics problems ranging from elementary algebra to more advanced word problems. The HCF is also critical in applications to find common denominators, simplify fractions, and allocate resources efficiently.
Related Links
LCM (Least Common Multiple) - Dive into our complete guide on LCM for Class 4 students with simple explanations and examples.
Math Calculator - Use our all-in-one Math Calculator to solve problems quickly
FAQs
1. What is the HCF of 36 and 24?
Answer: The HCF of 36 and 24 is 12.
Divisors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
Divisors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Common divisors: 1, 2, 3, 4, 6, 12
The highest common divisor is 12.
2. How to calculate HCF?
Answer: You can calculate the HCF by using any one of the following methods:
Prime Factorization: Factorize every number into its prime factors and take the lowest powers of common factors multiplied.
Division Method of HCF: Divide the bigger number by the smaller number and keep dividing the divisor by the remainder until the remainder becomes zero. The previous non-zero remainder is the HCF.
Listing Divisors: Take all the divisors of both the numbers and determine the greatest common divisor.
3. What is the HCF of 42 and 162?
Answer: The HCF of 42 and 162 is 6.
Divisors of 42: 1, 2, 3, 6, 7, 14, 21, 42
Divisors of 162: 1, 2, 3, 6, 9, 18, 27, 54, 81, 162
Common divisors: 1, 2, 3, 6
The highest common divisor is 6.
4. What is the HCF of 4 and 7?
Answer: The HCF of 4 and 7 is 1.
Divisors of 4: 1, 2, 4
Divisors of 7: 1, 7
The only shared divisor is 1, so 4 and 7 are co-prime.
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