How to Find the Least Common Multiple (LCM)

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A multiple of a number is the result of multiplying that number by a whole number. For example, multiples of \(6\) include \(6,12,18,24,30,\ldots\). A common multiple is a number that appears in the multiple lists for two or more numbers.

The least common multiple, or LCM, is the smallest positive common multiple. For example, multiples of \(8\) are \(8,16,24,32,40,\ldots\), and multiples of \(12\) are \(12,24,36,48,\ldots\). The first shared multiple is \(24\), so \(LCM(8,12)=24\).

Finding LCM with the GCF

For two positive whole numbers, the product of the numbers equals the product of their GCF and LCM:

\(a \times b = GCF(a,b) \times LCM(a,b)\)

So, \(LCM(a,b)=\dfrac{a \times b}{GCF(a,b)}\). This method is especially quick when the GCF is easy to find.

Example: \(GCF(18,24)=6\). Then \(LCM(18,24)=\dfrac{18 \times 24}{6}=72\). See the related GCF calculation here: Greatest Common Factor of 18 and 24.

Least Common Multiple

Think of this lesson as more than a rule to memorize. Least Common Multiple 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 Finding Least Common Multiple

1) \(LCM(6, \ 8) =\)

2) \(LCM(9, \ 12) =\)

3) \(LCM(10, \ 15) =\)

4) \(LCM(14, \ 21) =\)

5) \(LCM(16, \ 20) =\)

6) \(LCM(18, \ 24) =\)

7) \(LCM(25, \ 30) =\)

8) \(LCM(28, \ 42) =\)

9) \(LCM(32, \ 48) =\)

10) \(LCM(36, \ 45) =\)

11) \(LCM(40, \ 56) =\)

12) \(LCM(54, \ 72) =\)

13) \(LCM(63, \ 84) =\)

14) \(LCM(75, \ 90) =\)

15) \(LCM(96, \ 120) =\)

16) \(LCM(105, \ 140) =\)

17) \(LCM(121, \ 132) =\)

18) \(LCM(144, \ 180) =\)

19) \(LCM(168, \ 252) =\)

20) \(LCM(210, \ 330) =\)

Answers

1) \(LCM(6, \ 8) = \color{red}{24}\)

Solution:
Step 1: Find the GCF so the repeated factors are counted only once.
\(6 = 2 \times 3\)
\(8 = 2 \times 2 \times 2\)
So \(GCF(6, \ 8) = 2\).
Step 2: Use \(LCM(a,b)=\dfrac{a \times b}{GCF(a,b)}\).
\(LCM(6, \ 8)=\dfrac{6 \times 8}{2}\)
Step 3: Divide first, then multiply: \(6 \div 2 = 3\), and \(3 \times 8 = \color{red}{24}\).
Greatest Common Factor of 6 and 8

2) \(LCM(9, \ 12) = \color{red}{36}\)

Solution:
Step 1: Find the GCF so the repeated factors are counted only once.
\(9 = 3 \times 3\)
\(12 = 2 \times 2 \times 3\)
So \(GCF(9, \ 12) = 3\).
Step 2: Use \(LCM(a,b)=\dfrac{a \times b}{GCF(a,b)}\).
\(LCM(9, \ 12)=\dfrac{9 \times 12}{3}\)
Step 3: Divide first, then multiply: \(9 \div 3 = 3\), and \(3 \times 12 = \color{red}{36}\).
Greatest Common Factor of 9 and 12

3) \(LCM(10, \ 15) = \color{red}{30}\)

Solution:
Step 1: Find the GCF so the repeated factors are counted only once.
\(10 = 2 \times 5\)
\(15 = 3 \times 5\)
So \(GCF(10, \ 15) = 5\).
Step 2: Use \(LCM(a,b)=\dfrac{a \times b}{GCF(a,b)}\).
\(LCM(10, \ 15)=\dfrac{10 \times 15}{5}\)
Step 3: Divide first, then multiply: \(10 \div 5 = 2\), and \(2 \times 15 = \color{red}{30}\).
Greatest Common Factor of 10 and 15

4) \(LCM(14, \ 21) = \color{red}{42}\)

Solution:
Step 1: Find the GCF so the repeated factors are counted only once.
\(14 = 2 \times 7\)
\(21 = 3 \times 7\)
So \(GCF(14, \ 21) = 7\).
Step 2: Use \(LCM(a,b)=\dfrac{a \times b}{GCF(a,b)}\).
\(LCM(14, \ 21)=\dfrac{14 \times 21}{7}\)
Step 3: Divide first, then multiply: \(14 \div 7 = 2\), and \(2 \times 21 = \color{red}{42}\).
Greatest Common Factor of 14 and 21

5) \(LCM(16, \ 20) = \color{red}{80}\)

Solution:
Step 1: Find the GCF so the repeated factors are counted only once.
\(16 = 2 \times 2 \times 2 \times 2\)
\(20 = 2 \times 2 \times 5\)
So \(GCF(16, \ 20) = 4\).
Step 2: Use \(LCM(a,b)=\dfrac{a \times b}{GCF(a,b)}\).
\(LCM(16, \ 20)=\dfrac{16 \times 20}{4}\)
Step 3: Divide first, then multiply: \(16 \div 4 = 4\), and \(4 \times 20 = \color{red}{80}\).
Greatest Common Factor of 16 and 20

6) \(LCM(18, \ 24) = \color{red}{72}\)

Solution:
Step 1: Find the GCF so the repeated factors are counted only once.
\(18 = 2 \times 3 \times 3\)
\(24 = 2 \times 2 \times 2 \times 3\)
So \(GCF(18, \ 24) = 6\).
Step 2: Use \(LCM(a,b)=\dfrac{a \times b}{GCF(a,b)}\).
\(LCM(18, \ 24)=\dfrac{18 \times 24}{6}\)
Step 3: Divide first, then multiply: \(18 \div 6 = 3\), and \(3 \times 24 = \color{red}{72}\).
Greatest Common Factor of 18 and 24

7) \(LCM(25, \ 30) = \color{red}{150}\)

Solution:
Step 1: Find the GCF so the repeated factors are counted only once.
\(25 = 5 \times 5\)
\(30 = 2 \times 3 \times 5\)
So \(GCF(25, \ 30) = 5\).
Step 2: Use \(LCM(a,b)=\dfrac{a \times b}{GCF(a,b)}\).
\(LCM(25, \ 30)=\dfrac{25 \times 30}{5}\)
Step 3: Divide first, then multiply: \(25 \div 5 = 5\), and \(5 \times 30 = \color{red}{150}\).
Greatest Common Factor of 25 and 30

8) \(LCM(28, \ 42) = \color{red}{84}\)

Solution:
Step 1: Find the GCF so the repeated factors are counted only once.
\(28 = 2 \times 2 \times 7\)
\(42 = 2 \times 3 \times 7\)
So \(GCF(28, \ 42) = 14\).
Step 2: Use \(LCM(a,b)=\dfrac{a \times b}{GCF(a,b)}\).
\(LCM(28, \ 42)=\dfrac{28 \times 42}{14}\)
Step 3: Divide first, then multiply: \(28 \div 14 = 2\), and \(2 \times 42 = \color{red}{84}\).
Greatest Common Factor of 28 and 42

9) \(LCM(32, \ 48) = \color{red}{96}\)

Solution:
Step 1: Find the GCF so the repeated factors are counted only once.
\(32 = 2 \times 2 \times 2 \times 2 \times 2\)
\(48 = 2 \times 2 \times 2 \times 2 \times 3\)
So \(GCF(32, \ 48) = 16\).
Step 2: Use \(LCM(a,b)=\dfrac{a \times b}{GCF(a,b)}\).
\(LCM(32, \ 48)=\dfrac{32 \times 48}{16}\)
Step 3: Divide first, then multiply: \(32 \div 16 = 2\), and \(2 \times 48 = \color{red}{96}\).
Greatest Common Factor of 32 and 48

10) \(LCM(36, \ 45) = \color{red}{180}\)

Solution:
Step 1: Find the GCF so the repeated factors are counted only once.
\(36 = 2 \times 2 \times 3 \times 3\)
\(45 = 3 \times 3 \times 5\)
So \(GCF(36, \ 45) = 9\).
Step 2: Use \(LCM(a,b)=\dfrac{a \times b}{GCF(a,b)}\).
\(LCM(36, \ 45)=\dfrac{36 \times 45}{9}\)
Step 3: Divide first, then multiply: \(36 \div 9 = 4\), and \(4 \times 45 = \color{red}{180}\).
Greatest Common Factor of 36 and 45

11) \(LCM(40, \ 56) = \color{red}{280}\)

Solution:
Step 1: Find the GCF so the repeated factors are counted only once.
\(40 = 2 \times 2 \times 2 \times 5\)
\(56 = 2 \times 2 \times 2 \times 7\)
So \(GCF(40, \ 56) = 8\).
Step 2: Use \(LCM(a,b)=\dfrac{a \times b}{GCF(a,b)}\).
\(LCM(40, \ 56)=\dfrac{40 \times 56}{8}\)
Step 3: Divide first, then multiply: \(40 \div 8 = 5\), and \(5 \times 56 = \color{red}{280}\).
Greatest Common Factor of 40 and 56

12) \(LCM(54, \ 72) = \color{red}{216}\)

Solution:
Step 1: Find the GCF so the repeated factors are counted only once.
\(54 = 2 \times 3 \times 3 \times 3\)
\(72 = 2 \times 2 \times 2 \times 3 \times 3\)
So \(GCF(54, \ 72) = 18\).
Step 2: Use \(LCM(a,b)=\dfrac{a \times b}{GCF(a,b)}\).
\(LCM(54, \ 72)=\dfrac{54 \times 72}{18}\)
Step 3: Divide first, then multiply: \(54 \div 18 = 3\), and \(3 \times 72 = \color{red}{216}\).
Greatest Common Factor of 54 and 72

13) \(LCM(63, \ 84) = \color{red}{252}\)

Solution:
Step 1: Find the GCF so the repeated factors are counted only once.
\(63 = 3 \times 3 \times 7\)
\(84 = 2 \times 2 \times 3 \times 7\)
So \(GCF(63, \ 84) = 21\).
Step 2: Use \(LCM(a,b)=\dfrac{a \times b}{GCF(a,b)}\).
\(LCM(63, \ 84)=\dfrac{63 \times 84}{21}\)
Step 3: Divide first, then multiply: \(63 \div 21 = 3\), and \(3 \times 84 = \color{red}{252}\).
Greatest Common Factor of 63 and 84

14) \(LCM(75, \ 90) = \color{red}{450}\)

Solution:
Step 1: Find the GCF so the repeated factors are counted only once.
\(75 = 3 \times 5 \times 5\)
\(90 = 2 \times 3 \times 3 \times 5\)
So \(GCF(75, \ 90) = 15\).
Step 2: Use \(LCM(a,b)=\dfrac{a \times b}{GCF(a,b)}\).
\(LCM(75, \ 90)=\dfrac{75 \times 90}{15}\)
Step 3: Divide first, then multiply: \(75 \div 15 = 5\), and \(5 \times 90 = \color{red}{450}\).
Greatest Common Factor of 75 and 90

15) \(LCM(96, \ 120) = \color{red}{480}\)

Solution:
Step 1: Find the GCF so the repeated factors are counted only once.
\(96 = 2 \times 2 \times 2 \times 2 \times 2 \times 3\)
\(120 = 2 \times 2 \times 2 \times 3 \times 5\)
So \(GCF(96, \ 120) = 24\).
Step 2: Use \(LCM(a,b)=\dfrac{a \times b}{GCF(a,b)}\).
\(LCM(96, \ 120)=\dfrac{96 \times 120}{24}\)
Step 3: Divide first, then multiply: \(96 \div 24 = 4\), and \(4 \times 120 = \color{red}{480}\).
Greatest Common Factor of 96 and 120

16) \(LCM(105, \ 140) = \color{red}{420}\)

Solution:
Step 1: Find the GCF so the repeated factors are counted only once.
\(105 = 3 \times 5 \times 7\)
\(140 = 2 \times 2 \times 5 \times 7\)
So \(GCF(105, \ 140) = 35\).
Step 2: Use \(LCM(a,b)=\dfrac{a \times b}{GCF(a,b)}\).
\(LCM(105, \ 140)=\dfrac{105 \times 140}{35}\)
Step 3: Divide first, then multiply: \(105 \div 35 = 3\), and \(3 \times 140 = \color{red}{420}\).
Greatest Common Factor of 105 and 140

17) \(LCM(121, \ 132) = \color{red}{1452}\)

Solution:
Step 1: Find the GCF so the repeated factors are counted only once.
\(121 = 11 \times 11\)
\(132 = 2 \times 2 \times 3 \times 11\)
So \(GCF(121, \ 132) = 11\).
Step 2: Use \(LCM(a,b)=\dfrac{a \times b}{GCF(a,b)}\).
\(LCM(121, \ 132)=\dfrac{121 \times 132}{11}\)
Step 3: Divide first, then multiply: \(121 \div 11 = 11\), and \(11 \times 132 = \color{red}{1452}\).
Greatest Common Factor of 121 and 132

18) \(LCM(144, \ 180) = \color{red}{720}\)

Solution:
Step 1: Find the GCF so the repeated factors are counted only once.
\(144 = 2 \times 2 \times 2 \times 2 \times 3 \times 3\)
\(180 = 2 \times 2 \times 3 \times 3 \times 5\)
So \(GCF(144, \ 180) = 36\).
Step 2: Use \(LCM(a,b)=\dfrac{a \times b}{GCF(a,b)}\).
\(LCM(144, \ 180)=\dfrac{144 \times 180}{36}\)
Step 3: Divide first, then multiply: \(144 \div 36 = 4\), and \(4 \times 180 = \color{red}{720}\).
Greatest Common Factor of 144 and 180

19) \(LCM(168, \ 252) = \color{red}{504}\)

Solution:
Step 1: Find the GCF so the repeated factors are counted only once.
\(168 = 2 \times 2 \times 2 \times 3 \times 7\)
\(252 = 2 \times 2 \times 3 \times 3 \times 7\)
So \(GCF(168, \ 252) = 84\).
Step 2: Use \(LCM(a,b)=\dfrac{a \times b}{GCF(a,b)}\).
\(LCM(168, \ 252)=\dfrac{168 \times 252}{84}\)
Step 3: Divide first, then multiply: \(168 \div 84 = 2\), and \(2 \times 252 = \color{red}{504}\).
Greatest Common Factor of 168 and 252

20) \(LCM(210, \ 330) = \color{red}{2310}\)

Solution:
Step 1: Find the GCF so the repeated factors are counted only once.
\(210 = 2 \times 3 \times 5 \times 7\)
\(330 = 2 \times 3 \times 5 \times 11\)
So \(GCF(210, \ 330) = 30\).
Step 2: Use \(LCM(a,b)=\dfrac{a \times b}{GCF(a,b)}\).
\(LCM(210, \ 330)=\dfrac{210 \times 330}{30}\)
Step 3: Divide first, then multiply: \(210 \div 30 = 7\), and \(7 \times 330 = \color{red}{2310}\).
Greatest Common Factor of 210 and 330

How to Find the Least Common Multiple (LCM)