最大公因数计算器
为简化分数和分解因数求最大公因数
GCF Properties
calculators.gcf.whenToUseGCF
来源与方法
Euclidean algorithm: GCD(a,b) = GCD(b, a mod b)Greatest common factor/divisor
How to Use the GCF Calculator
The GCF (Greatest Common Factor) calculator finds the largest number that divides evenly into two or more numbers. Also known as GCD (Greatest Common Divisor), see solutions using prime factorization and the Euclidean algorithm.
What is GCF?
The Greatest Common Factor is the largest positive integer that divides all given numbers without a remainder:
GCF(12, 18) = 6 because 6 is the largest number that divides both 12 and 18.
Method 1: Prime Factorization
- Find the prime factorization of each number
- Identify common prime factors
- Take the lowest power of each common prime
- Multiply them together
Example: GCF(48, 36)
48 = 2⁴ × 3
36 = 2² × 3²
GCF = 2² × 3 = 12
Method 2: Euclidean Algorithm
Repeatedly divide and take remainders until you reach 0:
GCF(48, 18):
48 ÷ 18 = 2 remainder 12
18 ÷ 12 = 1 remainder 6
12 ÷ 6 = 2 remainder 0
GCF = 6
Method 3: Listing Factors
List all factors and find the largest common one:
Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 18: 1, 2, 3, 6, 9, 18
GCF = 6
Real-World Applications
- Simplifying fractions: Divide both parts by the GCF
- Tiling: Finding the largest square tile that fits
- Distributing items: Dividing evenly into groups
Related Calculators
For finding the LCM, use our LCM calculator. For simplifying fractions, try our fraction calculator.
常见问题
The Greatest Common Factor (GCF), also called GCD (Greatest Common Divisor), is the largest positive number that divides both numbers evenly. For example, GCF(12, 18) = 6 because 6 is the largest number that divides both (12÷6=2, 18÷6=3).
Divide both numerator and denominator by their GCF. For 12/18: GCF(12,18) = 6. Divide both by 6: 12÷6 / 18÷6 = 2/3. The fraction is now in lowest terms because the new GCF is 1.
A fast method to find GCF using repeated division. For GCF(48, 18): 48÷18 = 2 R12, then 18÷12 = 1 R6, then 12÷6 = 2 R0. When remainder is 0, the divisor (6) is the GCF. This works for any two numbers.
Numbers with GCF = 1 are called "relatively prime" or "coprime." They share no common factors besides 1. Examples: 8 and 15, 9 and 14. A fraction with coprime numerator and denominator is already fully simplified.
Find the GCF of the first two, then find the GCF of that result with the third, and so on. Or use prime factorization: the GCF contains only primes common to all numbers, each with its lowest power. GCF(12, 18, 24) = 6.
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