Precalculus Examples

$\frac{1}{3}$ , $2$ , $12$ , $72$

Step 1

To find the LCM for a list of fractions such as $LCM (\frac{a}{b}, \frac{c}{d})$ using GCF:

1. Find the LCM of $a$ and $c$ .

2. Find the GCF of $b$ and $d$ .

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

Step 2

Calculate the LCM of the numerators $1, 2, 12, 72$ .

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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 number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 2.3

Since $2$ has no factors besides $1$ and $2$ .

$2$ is a prime number

Step 2.4

The prime factors for $12$ are $2 \cdot 2 \cdot 3$ .

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Step 2.4.1

$12$ has factors of $2$ and $6$ .

$2 \cdot 6$

Step 2.4.2

$6$ has factors of $2$ and $3$ .

$2 \cdot 2 \cdot 3$

Step 2.5

The prime factors for $72$ are $2 \cdot 2 \cdot 2 \cdot 3 \cdot 3$ .

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Step 2.5.1

$72$ has factors of $2$ and $36$ .

$2 \cdot 36$

Step 2.5.2

$36$ has factors of $2$ and $18$ .

$2 \cdot 2 \cdot 18$

Step 2.5.3

$18$ has factors of $2$ and $9$ .

$2 \cdot 2 \cdot 2 \cdot 9$

Step 2.5.4

$9$ has factors of $3$ and $3$ .

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

Step 2.6

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

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Step 2.6.1

Multiply $2$ by $2$ .

$4 \cdot 2 \cdot 3 \cdot 3$

Step 2.6.2

Multiply $4$ by $2$ .

$8 \cdot 3 \cdot 3$

Step 2.6.3

Multiply $8$ by $3$ .

$24 \cdot 3$

Step 2.6.4

Multiply $24$ by $3$ .

$72$

Step 3

Find the GCF of the denominators $3, 1, 1, 1$ .

$GCF (3, 1, 1, 1) = 1$

Step 4

Divide $72$ by $1$ .

$72$