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How to Find the Common Denominator Using Factorization

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To use this method, you need to be comfortable with prime number factorization. This method works no matter what your denominators are.

Rule

Lowest Common Denominator, LCD

1.: Write down all the denominators below each other.
2.: Find the prime factors of each denominator.
3.: All the factors of the different denominators have to be included as the factors of the common denominator.
4.: The product of those factors is your lowest common denominator.

Example 1

Find $\frac{1}{16} + \frac{3}{20} + \frac{4}{30}$ .

First, you find the common denominator:


Denominator	Factors

$16$	$2 \cdot 2 \cdot 2 \cdot 2$

$20$	$2 \cdot 2 \cdot 5$

$30$	$2 \cdot 3 \cdot 5$

LCD	$2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 5$

The least common denominator is then:

2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 5 = 240

Then, you expand the fractions so all of them have the common denominator: $16$ is missing the factors $3$ and $5$ ; $20$ is missing the factors $2$ , $2$ and $3$ ; and $30$ is missing the factors $2$ , $2$ and $2$ to have all the factors in the LCD. That gives you

\begin{array}{l} \frac{1}{16} + \frac{3}{20} + \frac{4}{30} & = \frac{1 \cdot 3 \cdot 5}{16 \cdot 3 \cdot 5} + \frac{3 \cdot 2 \cdot 2 \cdot 3}{20 \cdot 2 \cdot 2 \cdot 3} \\ + \frac{4 \cdot 2 \cdot 2 \cdot 2}{30 \cdot 2 \cdot 2 \cdot 2} \\ = \frac{15}{240} + \frac{36}{240} + \frac{32}{240} \\ = \frac{15 + 36 + 32}{240} \\ = \frac{83}{240} . \end{array}

\begin{array}{l} \frac{1}{16} + \frac{3}{20} + \frac{4}{30} & = \frac{1 \cdot 3 \cdot 5}{16 \cdot 3 \cdot 5} + \frac{3 \cdot 2 \cdot 2 \cdot 3}{20 \cdot 2 \cdot 2 \cdot 3} + \frac{4 \cdot 2 \cdot 2 \cdot 2}{30 \cdot 2 \cdot 2 \cdot 2} \\ = \frac{15}{240} + \frac{36}{240} + \frac{32}{240} \\ = \frac{15 + 36 + 32}{240} \\ = \frac{83}{240} . \end{array}

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