Precalculus Examples

$\frac{4}{3}$ , $\frac{8}{3}$ , $\frac{16}{3}$ , $\frac{32}{3}$ , $\frac{64}{3}$

Step 1

To find the LCM for a list of fractions, check if denominators are similar or not.

Fractions with the same denominator:

1: $LCM (\frac{a}{b}, \frac{c}{b}) = \frac{LCM (a, c)}{b}$

Fractions with different denominators such as, $LCM (\frac{a}{b}, \frac{c}{d})$ :

1: Find the LCM of $b$ and $d = LCM (b, d)$

2: Multiply the numerator and denominator of the first fraction $\frac{a}{b}$ by $\frac{LCM (b, d)}{b}$

3: Multiply the numerator and denominator of the second fraction $\frac{c}{d}$ by $\frac{LCM (b, d)}{d}$

4: After making the denominators for all the fractions same, in this case, only two fractions, find the LCM of the new numerators

5: The LCM will be the $\frac{LCM (numerators)}{LCM (b, d)}$

Step 2

Since $\frac{4}{3}, \frac{8}{3}, \frac{16}{3}, \frac{32}{3}, \frac{64}{3}$ are all fractions with the same denominator, $LCM (\frac{4}{3}, \frac{8}{3}, \frac{16}{3}, \frac{32}{3}, \frac{64}{3}) = \frac{LCM (4, 8, 16, 32, 64)}{3}$ .

$LCM (\frac{4}{3}, \frac{8}{3}, \frac{16}{3}, \frac{32}{3}, \frac{64}{3}) = \frac{LCM (4, 8, 16, 32, 64)}{3}$

Step 3

Find the LCM for $4, 8, 16, 32, 64$ .

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Step 3.1

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 3.2

$4$ has factors of $2$ and $2$ .

$2 \cdot 2$

Step 3.3

The prime factors for $8$ are $2 \cdot 2 \cdot 2$ .

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Step 3.3.1

$8$ has factors of $2$ and $4$ .

$2 \cdot 4$

Step 3.3.2

$4$ has factors of $2$ and $2$ .

$2 \cdot 2 \cdot 2$

Step 3.4

The prime factors for $16$ are $2 \cdot 2 \cdot 2 \cdot 2$ .

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Step 3.4.1

$16$ has factors of $2$ and $8$ .

$2 \cdot 8$

Step 3.4.2

$8$ has factors of $2$ and $4$ .

$2 \cdot 2 \cdot 4$

Step 3.4.3

$4$ has factors of $2$ and $2$ .

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

Step 3.5

The prime factors for $32$ are $2 \cdot 2 \cdot 2 \cdot 2 \cdot 2$ .

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Step 3.5.1

$32$ has factors of $2$ and $16$ .

$2 \cdot 16$

Step 3.5.2

$16$ has factors of $2$ and $8$ .

$2 \cdot 2 \cdot 8$

Step 3.5.3

$8$ has factors of $2$ and $4$ .

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

Step 3.5.4

$4$ has factors of $2$ and $2$ .

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

Step 3.6

The prime factors for $64$ are $2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2$ .

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Step 3.6.1

$64$ has factors of $2$ and $32$ .

$2 \cdot 32$

Step 3.6.2

$32$ has factors of $2$ and $16$ .

$2 \cdot 2 \cdot 16$

Step 3.6.3

$16$ has factors of $2$ and $8$ .

$2 \cdot 2 \cdot 2 \cdot 8$

Step 3.6.4

$8$ has factors of $2$ and $4$ .

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

Step 3.6.5

$4$ has factors of $2$ and $2$ .

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

Step 3.7

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

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Step 3.7.1

Multiply $2$ by $2$ .

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

Step 3.7.2

Multiply $4$ by $2$ .

$8 \cdot 2 \cdot 2 \cdot 2$

Step 3.7.3

Multiply $8$ by $2$ .

$16 \cdot 2 \cdot 2$

Step 3.7.4

Multiply $16$ by $2$ .

$32 \cdot 2$

Step 3.7.5

Multiply $32$ by $2$ .

$64$

Step 4

The LCM for $\frac{4}{3}, \frac{8}{3}, \frac{16}{3}, \frac{32}{3}, \frac{64}{3}$ is the LCM of the numerator over the common denominator.

$\frac{64}{3}$