The Highest Common Factor (HCF), also known as the Greatest Common Divisor (GCD). It refers to the highest possible number that divides two given numbers without leaving a remainder. Finding the HCF is crucial in various mathematical operations and problem-solving scenarios.
There are several approaches to determine the HCF of two numbers, with one of the most efficient methods being the prime factorization technique.
Delve into the realm of HCF to explore its intricacies, properties, and applications. Gain insights into finding the HCF for a set of numbers, learn simplified techniques to compute it, and discover the HCF through division methods.
Embark on a fascinating journey through the realm of HCF, uncovering interesting facts and practical methods to compute this significant mathematical value.
The other names of Greatest Common Factor are
- Highest Common Factor (HCF)
- Highest Common Divisor (HCD)
- Greatest Common Divisor (GCD)
What is HCF?
HCF stands for Highest Common Factor. It refers to the largest number that divides two or more given numbers without leaving a remainder. It is also known as the Greatest Common Divisor (GCD).
Consider two numbers, 24 and 36. The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24. The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36.
To find the HCF, we look for the highest number that appears in the factor list of both 24 and 36. In this case, the highest common factor is 12, as it divides both 24 and 36 evenly without any remainder.
Therefore, the HCF of 24 and 36 is 12.
Example Of Greatest Common Factor
Example1: Find the Greatest Common Factor of 18 and 27
Solution
- Get all the factors of each number,
- Round the Common factors,
- Take the Greatest of those
How to Find HCF?
There are multiple methods to find the HCF (Highest Common Factor) of two or more numbers. Here are a few common approaches:
1. Prime Factorization:
Finding the HCF (Highest Common Factor) of two or more numbers using prime factorization is an effective method. Here’s how you can do it:
Step 1: Express each number as a product of its prime factors. For example, consider two numbers: 48 and 60.
- 48 = 24 * 31
- 60 = 22 * 31 * 51
Step 2: Identify the common prime factors of the given numbers. In this case, the common prime factors are 2 and 3.
Step 3: Multiply the common prime factors to obtain the HCF. HCF = 2 * 3 = 6
Therefore, the HCF of 48 and 60 is 6.
2. Division Method:
Finding the HCF (Highest Common Factor) of two or more numbers using the division method is another approach. Here’s a step-by-step guide:
Step 1: Start by selecting the two numbers for which you want to find the HCF. Let’s use the numbers 48 and 60 as an example.
Step 2: Divide the larger number by the smaller number.
- 60 ÷ 48 = 1 remainder 12
Step 3: Replace the larger number with the smaller number and the remainder with the smaller number.
- New numbers: 48, 12
Step 4: Repeat the division process until the remainder becomes zero.
- 48 ÷ 12 = 4 remainder 0
Step 5: The HCF is the last non-zero remainder obtained in the previous step, which is 12 in this case.
Therefore, the HCF of 48 and 60 is 12.
Using the division method, you can find the HCF of any set of numbers. It simplifies the process by repeatedly dividing the numbers until the remainder becomes zero. This method is efficient and widely used to find the HCF accurately.
3. Euclidean Algorithm:
- Divide the larger number by the smaller number.
- Find the remainder.
- Replace the larger number with the smaller number and the smaller number with the remainder.
- Repeat the process until the remainder becomes zero.
- The last non-zero remainder is the HCF.
Relation Between LCM and HCF
The HCF (Highest Common Factor) of two or more numbers represents their highest common factor, obtained by multiplying their common prime factors. On the other hand, the LCM (Least Common Multiple) of two or more numbers is the smallest number that is a multiple of all the given numbers.
The LCM (Least Common Multiple) and HCF (Highest Common Factor) of two or more numbers are closely related. Here is the relationship between LCM and HCF:
- Product Rule: The product of the LCM and HCF of two numbers is equal to the product of the original numbers.
- LCM(a, b) * HCF(a, b) = a * b
- Simplification Rule: If you have two numbers, a and b, and their LCM is L and HCF is H, you can simplify the numbers by dividing them both by their HCF.
- New numbers: a/H and b/H
- Their LCM becomes L/H.
HCF Examples
Example 1: Find the HCF of 36 and 48.
Method 1: Prime Factorization
Prime factorization of 36: 22 * 32
Prime factorization of 48: 2^4 * 31
Common prime factors: 22 * 31 = 12
Therefore, the HCF of 36 and 48 is 12.
Method 2: Division Method
Dividing 48 by 36:
48 ÷ 36 = 1 remainder 12
Dividing 36 by 12:
36 ÷ 12 = 3 remainder 0
Therefore, the HCF of 36 and 48 is 12.
Example 2: Find the HCF of 72 and 90.
Method 1: Prime Factorization
Prime factorization of 72: 23 * 32
Prime factorization of 90: 21 * 32 * 51
Common prime factors: 21 * 32 = 18
Therefore, the HCF of 72 and 90 is 18.
Method 2: Division Method
Dividing 90 by 72:
90 ÷ 72 = 1 remainder 18
Dividing 72 by 18:
72 ÷ 18 = 4 remainder 0
Therefore, the HCF of 72 and 90 is 18.
Example 3: Find the HCF of 54 and 72.
Method 1:
Prime Factorization Prime factorization of 54: 21 * 33
Prime factorization of 72: 23 * 32
Common prime factors: 21 * 32 = 18
Therefore, the HCF of 54 and 72 is 18.
Method 2:
Division Method Dividing 72 by 54: 72 ÷ 54 = 1 remainder 18
Dividing 54 by 18: 54 ÷ 18 = 3 remainder 0
Therefore, the HCF of 54 and 72 is 18.
Example 4: Find the HCF of 120 and 150.
Method 1:
Prime Factorization Prime factorization of 120: 23 * 31 * 51
Prime factorization of 150: 21 * 31 * 52
Common prime factors: 21 * 31 * 51 = 30
Therefore, the HCF of 120 and 150 is 30.
Method 2:
Division Method Dividing 150 by 120:
150 ÷ 120 = 1 remainder 30
Dividing 120 by 30: 120 ÷ 30 = 4 remainder 0
Therefore, the HCF of 120 and 150 is 30.
FAQs on HCF
Q1: What is the HCF of two prime numbers?
A1: The HCF of two prime numbers is always 1, as prime numbers have no common factors other than 1.
Q2: Can the HCF of two numbers be greater than both numbers?
A2: No, the HCF of two numbers cannot be greater than the numbers themselves. It will always be a factor of both numbers.
Q3: What is the HCF of three or more numbers?
A3: The HCF of three or more numbers is the largest number that divides all the given numbers without leaving any remainder.
Q4: Can the HCF of two numbers be zero?
A4: No, the HCF of two numbers cannot be zero. The HCF is always a positive integer.
Q5: How can I find the HCF of large numbers?
A5: For large numbers, it is recommended to use the prime factorization method or the division method to find their HCF efficiently.
Q6: What is the HCF of two consecutive numbers?
A6: The HCF of two consecutive numbers is always 1 since they do not have any common factors other than 1.
Q7: Is the HCF the same as the greatest common divisor (GCD)?
A7: Yes, the terms “HCF” (Highest Common Factor) and “GCD” (Greatest Common Divisor) are used interchangeably to refer to the same concept.
Q8: Can the HCF of two numbers be negative?
A8: No, the HCF of two numbers is always a positive integer.
Q9: Can we find the HCF of fractions?
A9: Yes, the HCF of fractions can be found by considering the numerators and denominators separately and finding their HCF.
Q10: What is the HCF of 0 and any number?
A10: The HCF of 0 and any number (except 0) is the number itself.