⚡ Quick Examples
What is the Greatest Common Factor (GCF)?
The Greatest Common Factor (GCF) — also called the Greatest Common Divisor (GCD) or Highest Common Factor (HCF) — is the largest positive integer that divides evenly into two or more numbers without leaving a remainder.
For example, the GCF of 12 and 18 is 6, because 6 is the largest number that divides both exactly.
Other Names for GCF
3 Methods to Find the GCF (with Examples)
Method 1: Listing Factors
List all factors of each number, then find the largest one they share.
Factors of 18: 1, 2, 3, 6, 9, 18
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Common factors: 1, 2, 3, 6 → GCF = 6
✔ Best for small numbers and quick visual checks.
Method 2: Prime Factorization
Break each number into prime factors, then multiply the shared primes using the lowest exponent.
48 = 2⁴ × 3
72 = 2³ × 3²
Shared (lowest powers): 2³ × 3¹ = 8 × 3 = GCF = 24
✔ Best for 3+ numbers or exam-style working.
Method 3: Euclidean Algorithm
Divide repeatedly, replacing with the remainder, until the remainder is 0.
252 ÷ 105 = 2 remainder 42
105 ÷ 42 = 2 remainder 21
42 ÷ 21 = 2 remainder 0
Remainder = 0 → GCF = 21
✔ Fastest method for very large numbers.
Real-World Uses of GCF
GCF isn't just homework — it solves practical everyday problems.
Dividing Things Equally
48 apples + 60 oranges. GCF(48,60) = 12 → make 12 identical bags, no leftovers.
Simplifying Fractions
Simplify 36/48: GCF = 12 → divide both → 3/4 in lowest terms.
Cutting Materials
Ropes of 90 cm & 126 cm. GCF = 18 cm is the longest cut with zero waste.
Screen Aspect Ratios
1920×1080: GCF = 120 → ratio 16:9, the universal widescreen standard.
GCF Quick-Reference Table
Common number pairs for instant lookup.
| Numbers | GCF | LCM | Best Method |
|---|---|---|---|
| 12, 18 | 6 | 36 | Listing factors |
| 24, 36 | 12 | 72 | Factoring |
| 48, 72 | 24 | 144 | Prime factorization |
| 100, 75 | 25 | 300 | Euclid's algorithm |
| 56, 98 | 14 | 392 | Prime factorization |
| 17, 31 | 1 | 527 | Coprime numbers |
| 144, 180, 216 | 36 | 1080 | Prime factorization |
| 1000, 750 | 250 | 3000 | Euclid's algorithm |
GCF Formulas & Properties
Find LCM when GCF is known, or vice versa.
Repeat until remainder = 0; last divisor is the GCF.
Result is the fraction in lowest terms.
Apply GCF pairwise and chain the results.
GCF Calculator: FAQs
What is the GCF of 0 and any number?
GCF(0, n) = n for any non-zero integer n, because every non-zero number divides 0. However, GCF(0, 0) is undefined.
What does a GCF of 1 mean?
When GCF = 1, the numbers are coprime (relatively prime). They share no factor other than 1. Example: GCF(8, 15) = 1.
What is the difference between GCF and LCM?
GCF is the largest number that divides all numbers evenly. LCM is the smallest number that all numbers divide into. For two numbers: GCF × LCM = a × b.
Can GCF be larger than either number?
No. GCF is always ≤ the smallest number. It equals the smallest number only when that number divides all others. e.g., GCF(6, 12, 18) = 6.
How do I find GCF of more than two numbers?
Apply GCF sequentially: GCF(a, b, c) = GCF(GCF(a, b), c). For example: GCF(12, 18, 24) = GCF(6, 24) = 6. Our calculator handles up to 10 numbers automatically.
How is GCF used to simplify fractions?
Divide both numerator and denominator by their GCF. Example: 36/48 → GCF(36,48) = 12 → 36÷12 / 48÷12 = 3/4.
What is the fastest method to find GCF?
The Euclidean algorithm is the fastest — O(log n) time complexity. Our calculator uses it internally for speed and handles numbers of any size accurately.
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