Least Common Multiple and Least Common Denominator

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The Least Common Denominator

The Least Common Multiple

A multiple of a number is a whole number times that number. For example, some multiples of 6 are

        6, 12, 18, 24, 30, and 36

If two numbers are given then a common multiple of the two numbers is a number that is a multiple of both. Of all the common multiples of two numbers, there is a smallest one which we call the least common multiple.

Example

Find the least common multiple of 6 and 9.

Solution

One way of solving this problem is to write out multiples of each and see what is common to the list:

        6, 12, 18, 24, 30, 36, ...        multiples of 6

        9, 18, 27, 36, 45, ...        multiples of 9

We see that the numbers 18 and 36 are both common multiples of 6 and 9. The least common multiple is the smallest which is 18.

Example

Find the least common multiple of 8 and 32.

Solution

Instead of listing many multiples of each, we just notice that 32 is a multiple of 8 and hence 32 is a common multiple. It is the first multiple of 32. We can conclude that 32 is the least common multiple of 8 and 32.

In general, the least common multiple of two numbers with one the multiple of the other is just the larger number.

Exercise
1. Find the least common multiple of 15 and 54.
2. Find the least common multiple of 9 and 81.
  
  As you saw from the Exercise A, writing out many multiples of each number can be tedious. There is an alternate method that may save time. The strategy is based on the following idea. A multiple of a number is a multiple of each of the prime divisors.
Steps in Finding the LCM
1. Write the prime factorization of each number
2. List the primes that occur in at least one of the factorizations
3. Form a product using each prime the greatest number of time it occurs in any one of the expressions
Example

Find the LCM of 45 and 21

Solution
1. 45 = 9 x 5 = 3 x 3 x 5
  21 = 3 x 7
2. 3, 5, and 7
3. 3 x 3 x 5 x 7 The prime 3 occurs two times as it does in 3 x 3 x 5
  = 9 x 5 x 7 = 45 x 7 = 315
Exercises

Find the LCM of
1. 18 and 40
2. 12 and 15
3. 27 and 10
The Least Common Denominator

We define the least common denominator of two fractions as the least common multiples of the denominators.

Examples

Find the least common denominator of
1. 3/4 and 9/10
2. 5/6 and 10/11
Solutions
1. We find the least common multiples of 4 and 10
  
  4 = 2 x 2 10 = 2 x 5
  
  So the least common denominator is
  
  2 x 2 x 5 = 20
2. We find the least common multiples of 6 and 11
  
  6 = 2 x 3 11 is prime
  
  So the least common denominator is
  
  2 x 3 x 11 = 66
Exercises

Find the least common denominator of
1. 3/14 and 2/63
2. 8/25 and 23/100
Building Up Fractions With a Least Common Denominator

We have already learned how to simplify a fraction by dividing through by a common factor. Sometimes it is convenient to be able to work this process in reverse.

Example

Build up the fraction to answer the question

        5           ?
              =
        6          24

Solution

We see that

        24 = 6 x 4

so

        5           5 x 4            20
              =                  =
        6           6 x 4            24

Exercise

Build up the fraction to answer the question

        3           ?
              =
        7          35

Example

Which number is larger:  5/8 or 9/14?

Solution

Since the denominators are different, these numbers are difficult to compare. Our strategy is to build up each fraction to fractions with the least common denominator. We first find the least common denominator:

        8 = 2 x 2 x 2        14 = 2 x 7

The least common denominator is

        2 x 2 x 2 x 7 = 56

The next step is to notice that

        8 x 7 = 56        and         14 x 4 = 56

We write

        5            5 x 7           35
               =                 =
        8            8 x 7           56

and

         9             9 x 4           36
                =                  =
        14           14 x 4          56

Since

        35           36
                <
        56           56

We conclude that

        5           9
              <
        8          14

Exercise

Which is larger:  3/10 or 7/25?

Example

Write the three fractions  1/6, 5/8 and 3/10 as equivalent fractions with the LCD as the denominators.

Solution

We have

        6 = 2 x 3         8 = 2 x 2 x 2        10 = 2 x 5

So the least common denominator is

        2 x 2 x 2 x 3 x 5 = 8 x 3 x 5 =   24 x 5 = 120

We write

        1           1 x 20            20
              =                   =
        6           6 x 20            120

        5           5 x 15            75
              =                    =
        8           8 x 15            120

        3             3 x 12              36
                =                     =
       10           10 x 12            120

Exercise

Write the three fractions  2/15, 4/9 and 3/25 as equivalent fractions with the LCD as the denominators.

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