Formula
For two positive integers a and b, repeatedly replace the larger number by its remainder after division until the remainder is 0; the last non-zero divisor is GCF(a,b). For three numbers, calculate GCF(GCF(a,b),c).
GCF Calculator
Find the greatest common factor of two or three whole numbers, with Euclidean-algorithm steps, factor-list checks, LCM relationship, classroom wording and a printable arithmetic worksheet record.
Worked example
For 24 and 36, divide 36 by 24 to get remainder 12, then divide 24 by 12 to get remainder 0. The last non-zero divisor is 12, so the GCF is 12. If 60 is added, GCF(12,60) remains 12.
Professional note
Master’s Tip: when simplifying fractions, print the original numerator and denominator beside the GCF. Dividing both by the same greatest factor proves the fraction was reduced without changing its value.
Regional and unit assumptions
Standard or basis: elementary number theory for positive integers. The calculator uses the Euclidean algorithm and labels GCF as the same quantity often called greatest common divisor (GCD) or highest common factor (HCF).
Assumptions and limitations
- Inputs are rounded to non-negative whole numbers because greatest common factor is defined here for integers, not decimals.
- A third number of 0 is treated as blank, so the calculator returns the GCF of the first two numbers only.
- The greatest common factor is positive for positive inputs and is never larger than the smallest entered positive number.
- The factor-list limit is only a classroom display aid; the Euclidean algorithm gives the exact result without listing every factor.
- Very large integers can exceed practical browser display precision, so exams, proofs and coding work should follow the required teacher, textbook or software standard.
Methodology & Accuracy
How this calculator is checked
CalculationTime pages are built around visible arithmetic: the formula, assumptions, worked example and practical limitations are shown so the result can be checked rather than simply trusted.
Formula used
For two positive integers a and b, repeatedly replace the larger number by its remainder after division until the remainder is 0; the last non-zero divisor is GCF(a,b). For three numbers, calculate GCF(GCF(a,b),c).
Standard or basis
Standard or basis: elementary number theory for positive integers. The calculator uses the Euclidean algorithm and labels GCF as the same quantity often called greatest common divisor (GCD) or highest common factor (HCF).
Where a calculator follows a named legal, trade or industry standard, that standard is cited visibly. Otherwise the page uses transparent general arithmetic and states its limits.Master's Tip
Master’s Tip: when simplifying fractions, print the original numerator and denominator beside the GCF. Dividing both by the same greatest factor proves the fraction was reduced without changing its value.
Related calculators
Questions
What is the GCF?
The greatest common factor is the largest positive whole number that divides every entered number evenly.
Is GCF the same as GCD or HCF?
Yes in ordinary arithmetic. GCF, greatest common divisor and highest common factor refer to the same largest shared factor.
How do I find the GCF of two numbers?
Use the Euclidean algorithm: divide, keep the remainder, then repeat until the remainder is 0. The last non-zero divisor is the GCF.
How do I find the GCF of three numbers?
Find the GCF of the first two numbers, then find the GCF of that result and the third number.
What should I print for a GCF worksheet?
Print the entered numbers, GCF result, Euclidean steps, shared-factor check, assumptions, page URL, date and room for student or teacher notes.
Calculation note
Greatest common factors make simplification auditable. They show the biggest shared divisor before a fraction is reduced, a ratio is simplified or a divisibility problem is explained.
GCF protects fraction simplification
When a fraction is reduced, the numerator and denominator must be divided by the same non-zero factor. Using the greatest common factor proves the result is fully simplified.
The Euclidean algorithm avoids long factor lists
Listing factors works for small numbers, but repeated division with remainders is faster, exact and easier to audit when numbers get larger.
GCF and LCM are partner ideas
The GCF divides the entered numbers; the least common multiple is divided by them. Showing the distinction helps students avoid mixing up factor and multiple problems.
Sources and further readingOpenStax Prealgebra — Multiples and factorsKhan Academy — Greatest common factor