Greatest Common Factor (GCF) — How to Find It

The greatest common factor (GCF) of two or more whole numbers is the largest positive whole number that divides each of them with no remainder. For $18$ and $24$ , the GCF is $6$ , because $6$ divides both exactly and no larger number does.

The formula, in words and symbols

There is no single algebraic formula like $a + b$ here; the GCF is defined by a condition. If we write the prime factorizations

m = \prod_i p_i^{a_i}, \qquad n = \prod_i p_i^{b_i},

then

\mathrm{GCF}(m,n) = \prod_i p_i^{\min(a_i,\,b_i)}.

In plain terms: for every prime the two numbers share, keep it raised to the smaller of its two exponents, then multiply those together. A factor divides a number exactly, a common factor is shared between the numbers, and the greatest common factor is the biggest shared one. For positive integers, GCF and greatest common divisor mean the same thing.

Why taking the smaller exponent works

A common factor can only contain primes that both numbers contain, otherwise it would fail to divide one of them. And it can use a shared prime only as many times as the poorer of the two numbers supplies it. Push the exponent one higher and the factor no longer divides the number with fewer copies of that prime. So the largest factor that still divides both is built from each shared prime at its minimum exponent. That is exactly the formula above.

Worked example: GCF of 18 and 24

Factor each number:

18 = 2 \cdot 3^2, \qquad 24 = 2^3 \cdot 3

Both share one $2$ and one $3$ . Taking the smaller exponent of each shared prime:

2^1 \cdot 3^1 = 6

\mathrm{GCF}(18,24) = 6.

You can confirm by listing factors too: $18$ has $1,2,3,6,9,18$ and $24$ has $1,2,3,4,6,8,12,24$ ; the largest in both lists is $6$ . Listing is quickest for small numbers, prime factorization stays clean for large ones.

Try these on your own

Find the GCF of $20$ and $30$ , first by listing factors, then by prime factorization; both methods should land on the same number. As a check, divide each original number by your answer and confirm you get whole numbers, with no larger candidate dividing both.

Calculation traps to avoid

Stopping too early. For $18$ and $24$ , both $2$ and $3$ are common, but neither alone is greatest.
Confusing factors with multiples. The GCF divides the values; it is not a number the values grow into.
Dropping a shared prime or using the wrong exponent during factorization.

Where the GCF earns its keep

A quick payoff is simplifying fractions. Dividing top and bottom of $\tfrac{18}{24}$ by their GCF $6$ gives

\frac{18}{24} = \frac{3}{4}.

The same tool finds the largest equal group size or the biggest unit that fits several quantities exactly.

Frequently Asked Questions

What is the greatest common factor?: The greatest common factor, or GCF, is the largest positive whole number that divides each of two or more whole numbers with no remainder. A factor divides a number exactly, a common factor is shared between numbers, and the greatest common factor is the biggest one they share. For positive integers, GCF and greatest common divisor usually mean the same thing.
How do you find the GCF of two numbers?: You can list all factors of each number and pick the largest one that appears in both lists, which is usually fastest for small numbers. For larger numbers, prime factorization is cleaner: break each number into prime factors, keep only the primes both numbers share, use the smaller exponent for each shared prime, and multiply those together.
What is the GCF of 18 and 24?: The GCF of 18 and 24 is 6. Using prime factorization, 18 is 2 times 3 squared and 24 is 2 cubed times 3. The shared primes with the smaller exponents are one 2 and one 3, and 2 times 3 is 6. Both 18 and 24 divide evenly by 6, and the next larger candidate, 12, does not divide 18.
What are common mistakes when finding the GCF?: One common mistake is stopping too early: for 18 and 24, both 2 and 3 are common factors, but neither is the greatest. Another is mixing up factors and multiples, since the GCF looks for numbers that divide both values, not numbers the values grow into. Students also lose shared prime factors or use the wrong exponent during factorization.
When do you use the greatest common factor?: GCF is especially useful for simplifying fractions, dividing items into the largest possible equal groups, and finding the biggest unit size that fits several quantities. Whenever a problem asks for the largest group size or the simplest form of a ratio or fraction, the greatest common factor is usually the tool to reach for.

Need help with a problem?

Upload your question and get a verified, step-by-step solution in seconds.

Open GPAI Solver →