Thinking Mathematically (6th Edition)

Published by Pearson
ISBN 10: 0321867327
ISBN 13: 978-0-32186-732-2

Chapter 5 - Number Theory and the Real Number System - 5.1 Number Theory: Prime and Composite Numbers - Exercise Set 5.1 - Page 257: 107


To find the least common multiple of two or more numbers, 1) Write the prime factorization of each number. 2) Select every prime factor that occurs, raised to the greatest power to which it occurs, in these factorization. 3) Form the product of the number from step 2. The least common multiple is the product of these factor. The least common multiple of two or more natural numbers can be obtained by finding out the smallest number that is divisible by both/all the natural numbers. For example, least common multiple of 48 and 72 is 144 and can be obtained as: Factors of 48 can be written as: \[\begin{align} & 48=2\times 2\times 2\times 3\times 2 \\ & ={{2}^{4}}\times 3 \end{align}\] and, factors of 72 can be written as: \[\begin{align} & 72=2\times 2\times 2\times 3\times 3 \\ & ={{2}^{3}}\times {{3}^{2}} \end{align}\] Now, take the greater exponents from both the numbers, that is, \[{{2}^{4}}\ \text{and }{{3}^{2}}\]. Thus, \[\begin{align} & \text{LCM=}{{\text{2}}^{4}}\times {{3}^{2}} \\ & =2\times 2\times 2\times 2\times 3\times 3 \\ & =144 \end{align}\] Hence, the least common multiple of 48 and 72 is 144.
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