How to Find the Greatest Common Factor (GCF & HCF): Step-by-Step Guide
The Greatest Common Factor (GCF), also widely known as the Highest Common Factor (HCF) or महत्तम समापवर्तक (म.स.), is the largest integer that divides two or more numbers without leaving any remainder. Finding the GCF is essential for simplifying fractions to lowest terms, factoring polynomials, and solving integer division word problems.
Method 1: Prime Factorization (Lowest Exponent Rule)
To compute GCF using prime factorization:
- Find the prime factors of each number.
- Identify only the common prime factors present in all numbers.
- Raise each common prime to its lowest exponent (power).
- Multiply these prime powers together.
Method 2: Euclidean Division Algorithm
The fastest method for large integers, based on Euclidean division:
- Divide the larger number by the smaller number to get quotient and remainder.
- Replace the larger number with the divisor, and the divisor with the remainder.
- Repeat the division until the remainder equals 0.
- The divisor at the step where remainder is 0 is the GCF / HCF.
Example: Finding GCF (HCF) of 18 and 24
Prime factorizations:
18 = 2¹ × 3²
24 = 2³ × 3¹
- Common prime factors: 2 and 3.
- Lowest power of 2:
2¹ = 2 - Lowest power of 3:
3¹ = 3 - GCF / HCF:
2 × 3 = 6(18 और 24 का महत्तम समापवर्तक 6 है).
Frequently Asked Questions (GCF / HCF FAQs)
What is the difference between GCF, HCF, and GCD?
They are identical mathematical terms. GCF stands for Greatest Common Factor (US), HCF stands for Highest Common Factor (India / UK / CBSE), and GCD stands for Greatest Common Divisor.
What if two numbers share no common prime factors?
If two positive integers share no prime factors in common, their GCF is 1. Such numbers are termed coprime or relatively prime (e.g. 15 and 28).
Can GCF ever be greater than the numbers?
No. The factor must divide into the numbers, so GCF is always less than or equal to the smallest number in the set.