## Thinking Mathematically (6th Edition)

Published by Pearson

# Chapter 5 - Number Theory and the Real Number System - Chapter 5 Test - Page 337: 3

#### Answer

GCD is $24$ and LCM is$144$.

#### Work Step by Step

Write 48 and 72 in terms of prime factors as follows: \begin{align} & 48=2\times 2\times 2\times 2\times 3 \\ & 72=2\times 2\times 2\times 3\times 3 \\ \end{align} Further express the factors in terms of exponents: \begin{align} & 48={{2}^{4}}\times 3 \\ & 72={{2}^{3}}\times {{3}^{2}} \\ \end{align} Now, for GCD select each prime factor with the smaller exponent that is common to each of the prime factorization: \begin{align} & \text{GCD}={{2}^{3}}\times 3 \\ & =8\times 3 \\ & =24 \end{align} Now, for LCM, select every prime factor that occurs, raised to the greater exponent in these prime factorizations: \begin{align} & \text{LCM}={{2}^{4}}\times {{3}^{2}} \\ & =16\times 9 \\ & =144 \end{align} Hence, the GCD is $24$ and LCM is$144$.

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