GCF calculator with Euclidean steps and reduced ratio
Paste 2 to 20 integers. HitEquals returns their greatest common factor, reduced ratio, common divisors and the successive Euclidean steps.
Inputs used by this calculator
Greatest Common Factor Calculator uses the fields below. The starting values reproduce the page example; replace them with values from the same units and time period before using the result.
| Input | Starting example | Accepted value | How it is used |
|---|---|---|---|
| Whole numbers separated by commas, spaces or lines | 84, 126, 210 | Text or a list in the format shown | Supplies the whole numbers separated by commas, spaces or lines quantity used by the published equation or result breakdown. |
What GCF means
The greatest common factor, also called greatest common divisor, is the largest positive integer that divides every input without remainder.
Euclidean algorithm
Divide and keep the remainder. Replace the pair with the divisor and remainder until the remainder is zero.
GCD(a,b)=GCD(b,a mod b)More than two integers
Find the GCF of the first pair, then combine that result with each remaining input.
Worked ratio example
84, 126 and 210 share 42. Dividing every term by 42 yields the simplest whole-number ratio 2:3:5.
Zeros and signs
Absolute values determine divisibility. GCD(a,0)=|a|, while an all-zero list has no positive greatest divisor in this tool.
Uses of a common factor
GCF reduces fractions and ratios, groups objects into equal largest sets and factors common coefficients from expressions.
GCF by listing factors versus Euclid
Listing factors is intuitive for small numbers, but the Euclidean algorithm is faster for large inputs because it repeatedly replaces a pair with the divisor and remainder. Both methods identify the same largest positive shared divisor.
GCF, LCM and fraction reduction
GCF divides quantities down to their simplest shared scale. LCM moves upward to the smallest shared multiple. To reduce a fraction, divide numerator and denominator by their GCF; to add fractions with different denominators, an LCM can supply a common denominator.
Euclidean example for 126 and 84
| Division | Quotient | Remainder |
|---|---|---|
| 126÷84 | 1 | 42 |
| 84÷42 | 2 | 0 |
| Result | GCF 42 |
GCF and LCM answer different questions
| Measure | Definition | Example for 12 and 18 |
|---|---|---|
| GCF | Largest shared divisor | 6 |
| LCM | Smallest positive shared multiple | 36 |
| Product check | GCF×LCM=|a×b| for two nonzero integers | 6×36=12×18 |
Frequently asked questions
Is GCF the same as GCD?
Yes for integers.
What is the GCF of 84 and 126?
It is 42.
Can the GCF be negative?
No. This page reports the conventional nonnegative divisor.
Can I include zero?
Yes, provided at least one number is nonzero.
How does GCF reduce a ratio?
Divide every term by the shared GCF.
Can I enter decimals?
No. GCF is defined here for whole integers.
What is the difference between GCF and LCM?
GCF is the largest common divisor; LCM is the smallest positive common multiple.
How does GCF simplify a fraction?
Divide both numerator and denominator by their GCF.
What to keep in mind
Signs are ignored and zeros are supported when at least one input is nonzero. Each absolute value must be at most one trillion; enter 2 to 20 integers.
Results are rounded for display in US format, with money in dollars and cents; CSV exports keep full calculation precision. This is a calculation summary, not an official certificate.