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 Find the least common multiple of 15 and 54. Hold mouse over the yellow rectangle for the solution  Find the least common multiple of 9 and 81. Hold mouse over the yellow rectangle for the solution  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 Write the prime factorization of each number List the primes that occur in at least one of the factorizations 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 45  =  9 x 5 =  3 x 3 x 5 21  =  3 x 7 3, 5, and 7 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    18 and 40 Hold mouse over the yellow rectangle for the solution  12 and 15 Hold mouse over the yellow rectangle for the solution  27 and 10 Hold mouse over the yellow rectangle for the solution    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  3/4 and 9/10 5/6 and 10/11 Solutions 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 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 3/14 and 2/63 Hold mouse over the yellow rectangle for the solution  8/25 and 23/100 Hold mouse over the yellow rectangle for the solution  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 Hold mouse over the yellow rectangle for the solution  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? Hold mouse over the yellow rectangle for the solution  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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