The formula this page uses
GCF(a, b) = the product of every prime both numbers share, taking the smaller power each time GCF(a, b) × LCM(a, b) = a × b
Greatest Common Factor Calculator
Use the free greatest common factor calculator to work through the number relationship, preserve exact values, and check the result with estimation. Choose a focused operation below to calculate, inspect the method, and connect the result to a visual model.
The visual updates with your calculation.
Result—Choose a mode, enter the known values, and calculate.
Show your work Step-by-step method
- The calculation method and verification will appear here.
Arithmetic
Greatest Common Factor Calculator explained
The short version
- The greatest common factor is the biggest number that divides two numbers with nothing left over.
- Every pair of numbers has one, because 1 always divides both, even when nothing bigger does.
- The GCF is what reduces a fraction to lowest terms in a single step.
What each part means
SymbolWhat it means
a and b — The two numbersWhole numbers above zero. The GCF is never bigger than the smaller one.
GCF — Greatest common factorAlso called the highest common factor or HCF. Same thing, different textbook.
prime — A building-block numberA number with no factors except 1 and itself: 2, 3, 5, 7, 11 and so on.
remainder — What is left after dividingIn the fast method, a remainder of 0 tells you to stop.
Show your work: a full example
- Find GCF(48, 60)48 and 60
- Break 48 into primes48 = 2 × 2 × 2 × 2 × 3
- Break 60 into primes60 = 2 × 2 × 3 × 5
- Keep the 2s they both have48 has four 2s and 60 has two, so keep 2 × 2 = 4
- Keep the 3s they both haveboth have exactly one 3, so keep 3
- Multiply the shared part4 × 3 = 12
- Check by dividing48 ÷ 12 = 4 and 60 ÷ 12 = 5, and 4 and 5 share nothing
A second, different case
- A different case: numbers too big to factorGCF(1071, 462)
- Divide and keep the remainder1071 ÷ 462 = 2 remainder 147
- Now divide the old bottom by that remainder462 ÷ 147 = 3 remainder 21
- Repeat147 ÷ 21 = 7 remainder 0
- The last remainder before 0 is the answerGCF(1071, 462) = 21
- Check both divisions1071 ÷ 21 = 51 and 462 ÷ 21 = 22
- Confirm nothing is left to share51 = 3 × 17 and 22 = 2 × 11, so no prime appears in both
Copy-ready example
48 and 60
Find GCF(48, 60)
Questions about the Greatest Common Factor Calculator
What is the quickest way for numbers in the hundreds?
Use the remainder chain. Divide the big number by the small one, keep the remainder, then divide the old small number by that remainder, and keep going until the remainder is 0. The last remainder you saw before the 0 is the answer, and it usually takes three or four lines.
How does the GCF reduce a fraction in one step?
Dividing the top and the bottom by the same number does not change the value, and the GCF is the biggest number that divides both without a remainder. So 36/60 with GCF 12 goes straight to 3/5, with no second round of tidying.
Can the GCF be one of the two numbers?
Yes, whenever the smaller number divides the bigger one. GCF(6, 24) = 6, because 6 divides itself and also divides 24. That is a useful sign that one number is a multiple of the other.
A word problem gives me two numbers. Do I want GCF or LCM?
If you are cutting something into equal groups or equal-sized pieces, you want the GCF, because the answer must divide both amounts. If you are waiting for two repeating events to line up again, you want the LCM, because the answer has to be a multiple of both.
Calculate, interpret, verify.
This workspace keeps the formula and the meaning together. Decimal results are rounded for display; retain full precision when you continue a calculation.
Enter known values
Match each input to the quantities in the problem and keep units consistent.
Use the relationship
The result panel identifies the formula or algorithm and shows the main substitutions.
Read the visual
Use the diagram, plot, or data display to check scale, direction, and plausibility.
Practice unaided
Move to targeted questions once you can explain why the method applies.
Learn the mathematics
Understand arithmetic behind this calculator
A calculator confirms an answer. Working the method yourself is what makes the next problem faster.