Examples

\frac{12}{5}

$\frac{12}{5}$ ,

\frac{9}{4}

$\frac{9}{4}$ ,

\frac{9}{2}

$\frac{9}{2}$

Step 1

To find the LCM for a list of fractions such as

LCM (\frac{a}{b}, \frac{c}{d})

$LCM (\frac{a}{b}, \frac{c}{d})$ using GCF:

1. Find the LCM of

a

$a$ and

c

$c$ .

2. Find the GCF of

b

$b$ and

d

$d$ .

LCM (\frac{a}{b}, \frac{c}{d}) = \frac{LCM (a, c)}{GCF (b, d)}

$LCM (\frac{a}{b}, \frac{c}{d}) = \frac{LCM (a, c)}{GCF (b, d)}$ .

Step 2

Calculate the LCM of the numerators

12, 9, 9

$12, 9, 9$ .

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Step 2.1

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 2.2

The prime factors for

12

$12$ are

2 \cdot 2 \cdot 3

$2 \cdot 2 \cdot 3$ .

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Step 2.2.1

12

$12$ has factors of

2

$2$ and

6

$6$ .

2 \cdot 6

$2 \cdot 6$

Step 2.2.2

6

$6$ has factors of

2

$2$ and

3

$3$ .

2 \cdot 2 \cdot 3

$2 \cdot 2 \cdot 3$

2 \cdot 2 \cdot 3

$2 \cdot 2 \cdot 3$

Step 2.3

9

$9$ has factors of

3

$3$ and

3

$3$ .

3 \cdot 3

$3 \cdot 3$

Step 2.4

The LCM of

12, 9, 9

$12, 9, 9$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

2 \cdot 2 \cdot 3 \cdot 3

$2 \cdot 2 \cdot 3 \cdot 3$

Step 2.5

Multiply

2 \cdot 2 \cdot 3 \cdot 3

$2 \cdot 2 \cdot 3 \cdot 3$ .

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Step 2.5.1

Multiply

2

$2$ by

2

$2$ .

4 \cdot 3 \cdot 3

$4 \cdot 3 \cdot 3$

Step 2.5.2

Multiply

4

$4$ by

3

$3$ .

12 \cdot 3

$12 \cdot 3$

Step 2.5.3

Multiply

12

$12$ by

3

$3$ .

36

$36$

36

$36$

36

$36$

Step 3

Find the GCF of the denominators

5, 4, 2

$5, 4, 2$ .

GCF (5, 4, 2) = 1

$GCF (5, 4, 2) = 1$

Step 4

Divide

36

$36$ by

1

$1$ .

36

$36$

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