The Highest Common Factor (HCF), also known as the greatest common divisor (GCD), is the largest number that divides two or more given numbers without leaving a remainder. It is a key concept in arithmetic, algebra, and number theory that simplifies calculations and problem-solving.
For example:
The HCF of 18 and 24 is 6 because 6 is the largest number that divides both 18 and 24 exactly.
HCF is particularly useful for:
Simplifying fractions to their lowest terms.
Solving problems involving ratios.
Dividing quantities into equal parts.
Manually calculating the HCF can be tedious, especially with large numbers. An HCF Calculator provides:
Quick and accurate results.
Solutions for multiple numbers at once.
Step-by-step explanations to understand the process.
This tool is ideal for students, teachers, and anyone working on arithmetic or algebraic problems.
While the calculator offers instant results, learning the manual methods builds mathematical intuition. Here are two common methods:
This method involves finding the prime factors of each number and identifying the common factors.
Example: Find the HCF of 18 and 24.
Prime factors of 18:
Prime factors of 24:
Common factors:
HCF:
This method involves repeated division until the remainder is 0.
Example: Find the HCF of 18 and 24.
Step 1: Divide the larger number by the smaller number:
remainder
Step 2: Divide the smaller number by the remainder:
remainder
HCF: The divisor when the remainder becomes 0 is .
Input Numbers: Enter the numbers for which you want to find the HCF.
Select Method (Optional): Choose the calculation method (e.g., prime factorization, division).
View Results: The calculator will display the HCF instantly and optionally provide step-by-step details.
Worked Examples on Highest Common Factor (HCF)
Five detailed step-by-step examples to help you understand how to find the HCF using different methods.
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Example 1: HCF of 18 and 24 (Prime Factorization Method)
1. Step 1: Find the prime factorization of each number:
– \( 18 = 2 \times 3^2 \)
– \( 24 = 2^3 \times 3 \)
2. Step 2: Identify the common factors:
– Common factors: \( 2 \) and \( 3 \)
3. Step 3: Multiply the smallest powers of the common factors:
– \( HCF = 2^1 \times 3^1 = 6 \)
4. Result: The HCF of 18 and 24 is 6.
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Example 2: HCF of 56 and 98 (Division Method)
1. Step 1: Divide the larger number by the smaller number:
– \( 98 \div 56 = 1 \), remainder \( 42 \)
2. Step 2: Divide the smaller number by the remainder:
– \( 56 \div 42 = 1 \), remainder \( 14 \)
3. Step 3: Divide the previous remainder by the new remainder:
– \( 42 \div 14 = 3 \), remainder \( 0 \)
4. Step 4: The divisor when the remainder becomes 0 is the HCF:
– \( HCF = 14 \)
5. Result: The HCF of 56 and 98 is 14.
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Example 3: HCF of 45 and 75 (Prime Factorization Method)
1. Step 1: Find the prime factorization of each number:
– \( 45 = 3^2 \times 5 \)
– \( 75 = 3 \times 5^2 \)
2. Step 2: Identify the common factors:
– Common factors: \( 3 \) and \( 5 \)
3. Step 3: Multiply the smallest powers of the common factors:
– \( HCF = 3^1 \times 5^1 = 15 \)
4. Result: The HCF of 45 and 75 is 15.
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Example 4: HCF of 36, 48, and 60 (Prime Factorization Method)
1. Step 1: Find the prime factorization of each number:
– \( 36 = 2^2 \times 3^2 \)
– \( 48 = 2^4 \times 3 \)
– \( 60 = 2^2 \times 3 \times 5 \)
2. Step 2: Identify the common factors:
– Common factor: \( 2^2 \) and \( 3^1 \)
3. Step 3: Multiply the smallest powers of the common factors:
– \( HCF = 2^2 \times 3^1 = 4 \times 3 = 12 \)
4. Result: The HCF of 36, 48, and 60 is 12.
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Example 5: HCF of 81 and 54 (Division Method)
1. Step 1: Divide the larger number by the smaller number:
– \( 81 \div 54 = 1 \), remainder \( 27 \)
2. Step 2: Divide the smaller number by the remainder:
– \( 54 \div 27 = 2 \), remainder \( 0 \)
3. Step 3: The divisor when the remainder becomes 0 is the HCF:
– \( HCF = 27 \)
4. Result: The HCF of 81 and 54 is 27.
