Compute the largest whole number that divides all inputs without remainder.
Overview
The greatest common factor (GCF), also called greatest common divisor (GCD), is the largest integer that divides every number in your set with no remainder. It is essential for simplifying fractions, reducing ratios, and solving problems about even grouping.
The Euclidean algorithm efficiently finds GCF by replacing (a,b) with (b, a mod b) until the remainder is zero. That method is faster than listing all factors for large integers and is why computer algebra systems rely on it.
When simplifying 42/70, divide numerator and denominator by GCF(42,70)=14 to get 3/5. The simplified fraction keeps the same value but uses smaller, easier numbers. The same idea applies to ratio reduction and repeating cycle alignment.
If you have more than two numbers, compute GCF pairwise: GCF(a,b,c) = GCF(GCF(a,b),c). This calculator handles that workflow so you do not miss a shared factor hidden across multiple values.
Steps
Type two or more whole numbers.
Apply the Euclidean algorithm internally.
Confirm the largest shared divisor.
Divide numerator and denominator by the GCF.
Verify each input divides evenly by the GCF.
Combine pairwise GCFs for larger sets.
Math
GCF(a,b) = GCF(b, a mod b), stop when remainder = 0
Each step preserves the common divisors of the pair. The last non-zero remainder is the greatest common factor.
Worked cases
126 mod 84 = 42; 84 mod 42 = 0
GCF = 42.
Divide both by GCF 14
3/5.
GCF(18,24)=6; GCF(6,30)=6
GCF = 6.
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Questions