Chapter 4.4, Problem 27WE

To determine

To find: the unknown integer.

Answer to Problem 27WE

Theunknown integer is $126$

Explanation of Solution

Given information:

The GCF of $84$ and another integer is $42$ , and their LCM is $252$

Calculation:

As per problem,

Let the other integer be x.

The GCF of $84$ and xis $42$ , so x is a multiple of $42$

Their LCM is $252$ , so x is factor of $252$

The multiples of $42$ which are also factors of $252$ are

  $42,84,\text{ and }126$

Now, LCM and GCF of $84$ and $84$ is $84$ , so x is not equal to $84$

And, the LCM and GCF of $84$ and $42$ is $84$ , so x is also not equal to $84$

However, LCM and GCF of $84$ and $126$ is

  $\begin{array}{l}84=2\cdot 42\\ 126=3\cdot 42\\ \text{LCM=}2\cdot 3\cdot 42=252\end{array}$

Therefore, $x=126$ , that is, the other integer is $126$