GCF Calculator
Find the greatest common factor (GCF/GCD) of two or more numbers
GCF Methods
What is the Greatest Common Factor?
The greatest common factor (GCF), also called the greatest common divisor (GCD) or highest common factor (HCF), is the largest positive integer that divides two or more numbers without leaving a remainder. For example, the GCF of 12 and 18 is 6.
GCF is fundamental in simplifying fractions, solving problems involving ratios, and number theory. When you reduce a fraction like 12/18 to 2/3, you're dividing both numerator and denominator by their GCF (6).
There are several methods to find GCF: listing all factors, prime factorization, or the efficient Euclidean algorithm. The Euclidean algorithm, dating back over 2,300 years, remains the fastest method for large numbers.
Methods to Find GCF
Listing Factors
List all factors of each number, find common ones, pick the largest.
Prime Factorization
Factor each number into primes, multiply common prime factors.
Euclidean Algorithm
Repeatedly apply GCF(a,b) = GCF(b, a mod b) until remainder is 0.
Division Method
Divide larger by smaller, use remainder to continue until zero.
Common GCF Examples
Reference table for frequently needed GCF calculations:
| Numbers | GCF | Common Factors | LCM |
|---|---|---|---|
| 12, 18 | 6 | 1, 2, 3, 6 | 36 |
| 24, 36 | 12 | 1, 2, 3, 4, 6, 12 | 72 |
| 15, 25 | 5 | 1, 5 | 75 |
| 48, 60 | 12 | 1, 2, 3, 4, 6, 12 | 240 |
| 100, 75 | 25 | 1, 5, 25 | 300 |
| 8, 12, 20 | 4 | 1, 2, 4 | 120 |
Applications of GCF
Simplifying Fractions
Divide numerator and denominator by GCF to reduce to lowest terms. 24/36 → divide by GCF(24,36)=12 → 2/3.
Tile Sizing
Find largest square tile that evenly covers a floor. For a 12×18 ft room, use 6×6 ft tiles (GCF = 6).
Equal Distribution
Split items evenly. With 24 apples and 36 oranges, you can make GCF(24,36)=6 identical gift bags.
Gear Ratios
Simplify gear ratios using GCF. A 48:32 ratio simplifies to 3:2 (divide by GCF=16).
Frequently Asked Questions
What's the difference between GCF and LCM?
GCF is the largest number that divides all given numbers. LCM is the smallest number divisible by all given numbers. They're related: GCF × LCM = product of the two numbers.
What if the numbers have no common factors?
Numbers with no common factors except 1 are called coprime or relatively prime. Their GCF is 1. Examples: 8 and 15, 9 and 14.
How do I find GCF of more than two numbers?
Find GCF of the first two numbers, then find GCF of that result with the third number, and so on. GCF(12, 18, 24) = GCF(GCF(12,18), 24) = GCF(6, 24) = 6.
Can GCF be larger than the numbers themselves?
No. GCF is always less than or equal to the smallest number. The GCF divides both numbers, so it can't exceed either of them.
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