How this instrument works
The greatest common divisor of three numbers extends the familiar two-number case by finding the GCD of the first pair, then taking the GCD of THAT result with the third number — the same shared-factor idea, just applied twice in a row. This works because any factor shared by all three numbers must, in particular, be shared by the first two, so folding them together first loses nothing.
This site's other GCD-related pages (its plain GCD, GCF, and Lowest Term calculators) cover the two-number case directly; this page extends that identical machinery to a third number, useful whenever a problem genuinely involves three quantities at once — splitting three different-length ropes into equal pieces, say, or finding the largest group size that divides three separate headcounts evenly.
Order doesn't matter: computing gcd(gcd(a,b),c) always gives the identical result as gcd(gcd(a,c),b) or gcd(gcd(b,c),a) — the greatest common divisor of three numbers is a single, well-defined value regardless of which pair is combined first.
- Enter the first number into the First number field.
- Enter the second and third numbers into their own fields.
- Read Greatest common divisor: the single largest number dividing all three evenly.
- Divide each of the three original numbers by that result and confirm every division comes out whole.
Worked example — 12, 18, and 30
For 12, 18, and 30: gcd(12,18)=6 first, and then gcd(6,30)=6 — so 6 is the largest number dividing all three evenly. Checking: 12÷6=2, 18÷6=3, and 30÷6=5, every division landing on a whole number.
7, 13, and 5 share no common factor at all beyond 1 — their combined GCD is 1, confirming the three numbers are collectively coprime. 100, 75, and 50 share a GCD of 25: gcd(100,75)=25, and gcd(25,50)=25 as well.
Questions
How do you find the GCD of three numbers?
Find the GCD of any two of them first, then find the GCD of that result with the remaining third number — the answer comes out identical no matter which pair is combined first.
Why does the order of combining the three numbers not matter?
Because the greatest common divisor of three numbers is a single, well-defined value shared by all three simultaneously — folding any two together first, then bringing in the third, always converges on that same shared value.
How is this different from this site's plain GCD or GCF calculators?
Those pages cover exactly two numbers; this page extends the identical shared-factor idea to a third number, useful whenever a problem genuinely involves three separate quantities at once.
What does it mean if the three-number GCD is 1?
That the three numbers are collectively coprime — no factor greater than 1 is shared by all three at once, even if some pairs among them individually share a larger factor.
Can this be extended to four or more numbers?
Yes — the identical pairwise-combining approach extends to any number of values, folding one more number in at a time; this page is scoped to three specifically to keep the calculation concrete and quick to check by hand.