How to Find the Greatest Common Factor (GCF)

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The greatest common factor, or GCF, is the largest whole number that divides two or more numbers evenly. For example, the factors of \(18\) are \(1,2,3,6,9,18\), and the factors of \(24\) are \(1,2,3,4,6,8,12,24\). The common factors are \(1,2,3,6\), so \(GCF(18,24)=6\).

GCF is useful when simplifying fractions because you divide the numerator and denominator by their greatest shared factor.

Prime Factor Method

Write each number as a product of prime factors.
Circle the prime factors that appear in both factorizations.
Multiply the shared prime factors. If there are no shared prime factors, the GCF is \(1\).

Example: \(72 = 2 \times 2 \times 2 \times 3 \times 3\) and \(60 = 2 \times 2 \times 3 \times 5\). The shared factors are \(2,2,\) and \(3\), so \(GCF(72,60)=2 \times 2 \times 3=12\). You can also check examples such as Greatest Common Factor of 72 and 60.

Find the Greatest Common Factor Video

Greatest Common Factor

Think of this lesson as more than a rule to memorize. Greatest Common Factor is about number sense, equivalent forms, and careful arithmetic. A strong student does not rush to the first formula on the page; they pause, identify the structure of the problem, and then choose the tool that matches that structure. That pause is what prevents most mistakes.

Fractions compare a part to a whole. Keep track of the numerator, denominator, and whether the pieces are the same size before adding, subtracting, or simplifying.

Here is the teacher way to approach the topic. First, read the problem slowly and underline the information that is actually given. Next, name the target: are you finding a value, simplifying an expression, comparing two quantities, solving for a variable, or interpreting a graph? Once the target is clear, the calculation becomes much less mysterious because every step has a job.

Read what is given and what is being asked.
Choose the rule that connects them.
Substitute carefully and simplify in small steps.
Check the final answer against the original question.

A helpful habit is to say what each number represents before using it. For example, if a number is a denominator, a radius, a slope, a common difference, or a coefficient, it should not be treated like an ordinary loose number. Its role tells you where it belongs in the formula. This is especially important on ACT-style questions because many wrong answer choices come from using the right numbers in the wrong places.

Another good habit is to keep the work organized vertically. Write one clean line for substitution, one line for simplifying, and one line for the final answer. If the problem has units, keep the units attached. If the problem has signs, exponents, or parentheses, copy them carefully from one line to the next. Most errors in this topic are not caused by a hard idea; they are caused by dropping a negative sign, combining unlike terms, using the wrong denominator, or skipping a check.

When you finish, ask a quick reasonableness question. Should the answer be positive or negative? Should it be larger or smaller than the original number? Does it fit the graph, table, shape, or equation? Can you plug it back into the original problem? This final check turns the lesson from a procedure into understanding.

On a test, the goal is not to write the longest solution. The goal is to write enough clear work that you can see the structure, avoid traps, and recover quickly if one line goes wrong. Practice the examples below with that mindset: identify the type of problem, choose the matching rule, show the substitution, simplify carefully, and check the answer before moving on.

Free printable Worksheets

Exercises for Greatest Common Factor

1) \(GCF(12, \ 18) =\)

2) \(GCF(20, \ 28) =\)

3) \(GCF(27, \ 45) =\)

4) \(GCF(32, \ 48) =\)

5) \(GCF(35, \ 64) =\)

6) \(GCF(42, \ 70) =\)

7) \(GCF(54, \ 81) =\)

8) \(GCF(72, \ 96) =\)

9) \(GCF(84, \ 126) =\)

10) \(GCF(90, \ 150) =\)

11) \(GCF(96, \ 144) =\)

12) \(GCF(108, \ 180) =\)

13) \(GCF(121, \ 187) =\)

14) \(GCF(132, \ 198) =\)

15) \(GCF(168, \ 252) =\)

16) \(GCF(210, \ 315) =\)

17) \(GCF(225, \ 360) =\)

18) \(GCF(256, \ 320) =\)

19) \(GCF(378, \ 462) =\)

20) \(GCF(540, \ 756) =\)

Answers

1) \(GCF(12, \ 18) = \color{red}{6}\)

Solution:
Step 1: Prime factorize each number.
\(12 = 2 \times 2 \times 3\)
\(18 = 2 \times 3 \times 3\)
Step 2: Match only the prime factors that appear in both lists. The shared factors are \(2 \times 3\).
Step 3: Multiply the shared factors: \(2 \times 3 = \color{red}{6}\).
Greatest Common Factor of 12 and 18

2) \(GCF(20, \ 28) = \color{red}{4}\)

Solution:
Step 1: Prime factorize each number.
\(20 = 2 \times 2 \times 5\)
\(28 = 2 \times 2 \times 7\)
Step 2: Match only the prime factors that appear in both lists. The shared factors are \(2 \times 2\).
Step 3: Multiply the shared factors: \(2 \times 2 = \color{red}{4}\).
Greatest Common Factor of 20 and 28

3) \(GCF(27, \ 45) = \color{red}{9}\)

Solution:
Step 1: Prime factorize each number.
\(27 = 3 \times 3 \times 3\)
\(45 = 3 \times 3 \times 5\)
Step 2: Match only the prime factors that appear in both lists. The shared factors are \(3 \times 3\).
Step 3: Multiply the shared factors: \(3 \times 3 = \color{red}{9}\).
Greatest Common Factor of 27 and 45

4) \(GCF(32, \ 48) = \color{red}{16}\)

Solution:
Step 1: Prime factorize each number.
\(32 = 2 \times 2 \times 2 \times 2 \times 2\)
\(48 = 2 \times 2 \times 2 \times 2 \times 3\)
Step 2: Match only the prime factors that appear in both lists. The shared factors are \(2 \times 2 \times 2 \times 2\).
Step 3: Multiply the shared factors: \(2 \times 2 \times 2 \times 2 = \color{red}{16}\).
Greatest Common Factor of 32 and 48

5) \(GCF(35, \ 64) = \color{red}{1}\)

Solution:
Step 1: Prime factorize each number.
\(35 = 5 \times 7\)
\(64 = 2 \times 2 \times 2 \times 2 \times 2 \times 2\)
Step 2: Match only the prime factors that appear in both lists. The shared factors are \(1\).
Step 3: Multiply the shared factors: \(1 = \color{red}{1}\).
Greatest Common Factor of 35 and 64

6) \(GCF(42, \ 70) = \color{red}{14}\)

Solution:
Step 1: Prime factorize each number.
\(42 = 2 \times 3 \times 7\)
\(70 = 2 \times 5 \times 7\)
Step 2: Match only the prime factors that appear in both lists. The shared factors are \(2 \times 7\).
Step 3: Multiply the shared factors: \(2 \times 7 = \color{red}{14}\).
Greatest Common Factor of 42 and 70

7) \(GCF(54, \ 81) = \color{red}{27}\)

Solution:
Step 1: Prime factorize each number.
\(54 = 2 \times 3 \times 3 \times 3\)
\(81 = 3 \times 3 \times 3 \times 3\)
Step 2: Match only the prime factors that appear in both lists. The shared factors are \(3 \times 3 \times 3\).
Step 3: Multiply the shared factors: \(3 \times 3 \times 3 = \color{red}{27}\).
Greatest Common Factor of 54 and 81

8) \(GCF(72, \ 96) = \color{red}{24}\)

Solution:
Step 1: Prime factorize each number.
\(72 = 2 \times 2 \times 2 \times 3 \times 3\)
\(96 = 2 \times 2 \times 2 \times 2 \times 2 \times 3\)
Step 2: Match only the prime factors that appear in both lists. The shared factors are \(2 \times 2 \times 2 \times 3\).
Step 3: Multiply the shared factors: \(2 \times 2 \times 2 \times 3 = \color{red}{24}\).
Greatest Common Factor of 72 and 96

9) \(GCF(84, \ 126) = \color{red}{42}\)

Solution:
Step 1: Prime factorize each number.
\(84 = 2 \times 2 \times 3 \times 7\)
\(126 = 2 \times 3 \times 3 \times 7\)
Step 2: Match only the prime factors that appear in both lists. The shared factors are \(2 \times 3 \times 7\).
Step 3: Multiply the shared factors: \(2 \times 3 \times 7 = \color{red}{42}\).
Greatest Common Factor of 84 and 126

10) \(GCF(90, \ 150) = \color{red}{30}\)

Solution:
Step 1: Prime factorize each number.
\(90 = 2 \times 3 \times 3 \times 5\)
\(150 = 2 \times 3 \times 5 \times 5\)
Step 2: Match only the prime factors that appear in both lists. The shared factors are \(2 \times 3 \times 5\).
Step 3: Multiply the shared factors: \(2 \times 3 \times 5 = \color{red}{30}\).
Greatest Common Factor of 90 and 150

11) \(GCF(96, \ 144) = \color{red}{48}\)

Solution:
Step 1: Prime factorize each number.
\(96 = 2 \times 2 \times 2 \times 2 \times 2 \times 3\)
\(144 = 2 \times 2 \times 2 \times 2 \times 3 \times 3\)
Step 2: Match only the prime factors that appear in both lists. The shared factors are \(2 \times 2 \times 2 \times 2 \times 3\).
Step 3: Multiply the shared factors: \(2 \times 2 \times 2 \times 2 \times 3 = \color{red}{48}\).
Greatest Common Factor of 96 and 144

12) \(GCF(108, \ 180) = \color{red}{36}\)

Solution:
Step 1: Prime factorize each number.
\(108 = 2 \times 2 \times 3 \times 3 \times 3\)
\(180 = 2 \times 2 \times 3 \times 3 \times 5\)
Step 2: Match only the prime factors that appear in both lists. The shared factors are \(2 \times 2 \times 3 \times 3\).
Step 3: Multiply the shared factors: \(2 \times 2 \times 3 \times 3 = \color{red}{36}\).
Greatest Common Factor of 108 and 180

13) \(GCF(121, \ 187) = \color{red}{11}\)

Solution:
Step 1: Prime factorize each number.
\(121 = 11 \times 11\)
\(187 = 11 \times 17\)
Step 2: Match only the prime factors that appear in both lists. The shared factors are \(11\).
Step 3: Multiply the shared factors: \(11 = \color{red}{11}\).
Greatest Common Factor of 121 and 187

14) \(GCF(132, \ 198) = \color{red}{66}\)

Solution:
Step 1: Prime factorize each number.
\(132 = 2 \times 2 \times 3 \times 11\)
\(198 = 2 \times 3 \times 3 \times 11\)
Step 2: Match only the prime factors that appear in both lists. The shared factors are \(2 \times 3 \times 11\).
Step 3: Multiply the shared factors: \(2 \times 3 \times 11 = \color{red}{66}\).
Greatest Common Factor of 132 and 198

15) \(GCF(168, \ 252) = \color{red}{84}\)

Solution:
Step 1: Prime factorize each number.
\(168 = 2 \times 2 \times 2 \times 3 \times 7\)
\(252 = 2 \times 2 \times 3 \times 3 \times 7\)
Step 2: Match only the prime factors that appear in both lists. The shared factors are \(2 \times 2 \times 3 \times 7\).
Step 3: Multiply the shared factors: \(2 \times 2 \times 3 \times 7 = \color{red}{84}\).
Greatest Common Factor of 168 and 252

16) \(GCF(210, \ 315) = \color{red}{105}\)

Solution:
Step 1: Prime factorize each number.
\(210 = 2 \times 3 \times 5 \times 7\)
\(315 = 3 \times 3 \times 5 \times 7\)
Step 2: Match only the prime factors that appear in both lists. The shared factors are \(3 \times 5 \times 7\).
Step 3: Multiply the shared factors: \(3 \times 5 \times 7 = \color{red}{105}\).
Greatest Common Factor of 210 and 315

17) \(GCF(225, \ 360) = \color{red}{45}\)

Solution:
Step 1: Prime factorize each number.
\(225 = 3 \times 3 \times 5 \times 5\)
\(360 = 2 \times 2 \times 2 \times 3 \times 3 \times 5\)
Step 2: Match only the prime factors that appear in both lists. The shared factors are \(3 \times 3 \times 5\).
Step 3: Multiply the shared factors: \(3 \times 3 \times 5 = \color{red}{45}\).
Greatest Common Factor of 225 and 360

18) \(GCF(256, \ 320) = \color{red}{64}\)

Solution:
Step 1: Prime factorize each number.
\(256 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2\)
\(320 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 5\)
Step 2: Match only the prime factors that appear in both lists. The shared factors are \(2 \times 2 \times 2 \times 2 \times 2 \times 2\).
Step 3: Multiply the shared factors: \(2 \times 2 \times 2 \times 2 \times 2 \times 2 = \color{red}{64}\).
Greatest Common Factor of 256 and 320

19) \(GCF(378, \ 462) = \color{red}{42}\)

Solution:
Step 1: Prime factorize each number.
\(378 = 2 \times 3 \times 3 \times 3 \times 7\)
\(462 = 2 \times 3 \times 7 \times 11\)
Step 2: Match only the prime factors that appear in both lists. The shared factors are \(2 \times 3 \times 7\).
Step 3: Multiply the shared factors: \(2 \times 3 \times 7 = \color{red}{42}\).
Greatest Common Factor of 378 and 462

20) \(GCF(540, \ 756) = \color{red}{108}\)

Solution:
Step 1: Prime factorize each number.
\(540 = 2 \times 2 \times 3 \times 3 \times 3 \times 5\)
\(756 = 2 \times 2 \times 3 \times 3 \times 3 \times 7\)
Step 2: Match only the prime factors that appear in both lists. The shared factors are \(2 \times 2 \times 3 \times 3 \times 3\).
Step 3: Multiply the shared factors: \(2 \times 2 \times 3 \times 3 \times 3 = \color{red}{108}\).
Greatest Common Factor of 540 and 756