Greatest Common Factor (GCF) — How to Find It

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

You can find GCF by listing factors or by using prime factorization. Listing is usually fastest for small numbers. Prime factorization is usually cleaner when the numbers are larger.

Greatest Common Factor Meaning

A factor is a whole number that divides another whole number exactly. A common factor is a factor the numbers share. The greatest common factor is the biggest one they share.

That is why GCF shows up in grouping problems and in simplifying fractions. In many school settings, GCF and greatest common divisor mean the same thing for positive integers.

How To Find GCF

1. List The Factors

Write all factors of each number, then look for the largest one that appears in both lists.

For $18$, the factors are:

$$1,\ 2,\ 3,\ 6,\ 9,\ 18$$

For $24$, the factors are:

$$1,\ 2,\ 3,\ 4,\ 6,\ 8,\ 12,\ 24$$

The greatest factor in both lists is $6$.

2. Use Prime Factorization

Break each number into prime factors, then keep only the prime factors both numbers share. If a shared prime appears more than once, use the smaller exponent. That shared product is the GCF.

Worked Example: GCF Of 18 And 24

Find the GCF of $18$ and $24$ using prime factorization.

First factor each number:

$$18 = 2 \cdot 3^2$$ $$24 = 2^3 \cdot 3$$

Now keep only the primes both numbers share, using the smaller exponent for each shared prime. Both numbers have one $2$ in common and one $3$ in common:

$$2^1 \cdot 3^1 = 6$$

So:

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

A quick check confirms it. Both $18 \div 6$ and $24 \div 6$ are whole numbers, and the next larger candidate, $12$, does not divide $18$.

Common GCF Mistakes

One common mistake is stopping too early. For $18$ and $24$, both $2$ and $3$ are common factors, but neither is the greatest one.

Another mistake is mixing up factors and multiples. The GCF looks for numbers that divide both values exactly. It does not look for numbers the original values can grow into.

Students also sometimes lose shared prime factors when using factorization. If a prime appears in both numbers, it belongs in the GCF, but only up to the smaller exponent.

When You Use The Greatest Common Factor

GCF is especially useful when you want to simplify fractions, divide items into the largest equal groups, or find the biggest unit size that fits several measurements exactly.

For example, simplifying $\frac{18}{24}$ starts by dividing top and bottom by their GCF, which is $6$:

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

Try A Similar Problem

Try finding the GCF of $20$ and $30$ first by listing factors, then by prime factorization. If both methods give the same answer, the idea has clicked.