Examples

\frac{7}{2}

$\frac{7}{2}$ ,

\frac{45}{4}

$\frac{45}{4}$ ,

\frac{6}{13}

$\frac{6}{13}$

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

7, 45, 6

$7, 45, 6$ .

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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

Since

7

$7$ has no factors besides

1

$1$ and

7

$7$ .

7

$7$ is a prime number

Step 2.3

The prime factors for

45

$45$ are

3 \cdot 3 \cdot 5

$3 \cdot 3 \cdot 5$ .

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Step 2.3.1

45

$45$ has factors of

3

$3$ and

15

$15$ .

3 \cdot 15

$3 \cdot 15$

Step 2.3.2

15

$15$ has factors of

3

$3$ and

5

$5$ .

3 \cdot 3 \cdot 5

$3 \cdot 3 \cdot 5$

3 \cdot 3 \cdot 5

$3 \cdot 3 \cdot 5$

Step 2.4

6

$6$ has factors of

2

$2$ and

3

$3$ .

2 \cdot 3

$2 \cdot 3$

Step 2.5

The LCM of

7, 45, 6

$7, 45, 6$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

2 \cdot 3 \cdot 3 \cdot 5 \cdot 7

$2 \cdot 3 \cdot 3 \cdot 5 \cdot 7$

Step 2.6

Multiply

2 \cdot 3 \cdot 3 \cdot 5 \cdot 7

$2 \cdot 3 \cdot 3 \cdot 5 \cdot 7$ .

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Step 2.6.1

Multiply

2

$2$ by

3

$3$ .

6 \cdot 3 \cdot 5 \cdot 7

$6 \cdot 3 \cdot 5 \cdot 7$

Step 2.6.2

Multiply

6

$6$ by

3

$3$ .

18 \cdot 5 \cdot 7

$18 \cdot 5 \cdot 7$

Step 2.6.3

Multiply

18

$18$ by

5

$5$ .

90 \cdot 7

$90 \cdot 7$

Step 2.6.4

Multiply

90

$90$ by

7

$7$ .

630

$630$

630

$630$

630

$630$

Step 3

Find the GCF of the denominators

2, 4, 13

$2, 4, 13$ .

GCF (2, 4, 13) = 1

$GCF (2, 4, 13) = 1$

Step 4

Divide

630

$630$ by

1

$1$ .

630

$630$

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