Chapter 4.4, Problem 25WE

(a).

To determine

To find: the GCF of the given monomials.

Answer to Problem 25WE

The GCF of $26p^3{q}^2{r}^2,\text{ 39}{p}^2{q}^3{r}^2,$ and $\text{78}{p}^2{q}^2{r}^3$ is $13p^2{q}^2{r}^2$ .

Explanation of Solution

Given information:

The given monomials:

  $26p^3{q}^2{r}^2,\text{ 39}{p}^2{q}^3{r}^2,\text{ 78}{p}^2{q}^2{r}^3$

Calculation:

The prime factorization of the coefficients are:

  $\begin{array}{l}26=2\cdot 13\\ 39=3\cdot 13\\ 78=2\cdot 3\cdot 13\end{array}$

The GCF of $26,\text{ 39}$ and $78$ is $13$

Compare the powers of each variable occurring in all three monomials. Use the power with the least exponent.

Compare $p^3$ and $p^2$ . Use $p^2$ .

Compare $q^2$ and $q^3$ . Use $q^2$ .

Compare $r^2$ and $r^3$ . Use $r^2$ .

  $\text{GCF}=13\cdot p^2\cdot q^2\cdot r^2=13p^2{q}^2{r}^2$

Therefore, the GCF of $26p^3{q}^2{r}^2,\text{ 39}{p}^2{q}^3{r}^2,$ and $\text{78}{p}^2{q}^2{r}^3$ is $13p^2{q}^2{r}^2$ .

(b).

To determine

The LCM of the given monomials.

Answer to Problem 25WE

The LCM of $26p^3{q}^2{r}^2,\text{ 39}{p}^2{q}^3{r}^2,$ and $\text{78}{p}^2{q}^2{r}^3$ is $78p^3{q}^3{r}^3$ .

Explanation of Solution

Given information:

The given monomials:

  $26p^3{q}^2{r}^2,\text{ 39}{p}^2{q}^3{r}^2,\text{ 78}{p}^2{q}^2{r}^3$

Calculation:

As per problem,

The prime factorization of the coefficients are:

  $\begin{array}{l}26=2\cdot 13\\ 39=3\cdot 13\\ 78=2\cdot 3\cdot 13\end{array}$

The LCM of $26,\text{ 39}$ and $78$ is

  $2\cdot 3\cdot 13=78$

Compare the powers of each variable occurring in either monomial. Use the power with the greatest exponent.

Compare $p^3$ and $p^2$ . Use $p^3$ .

Compare $q^2$ and $q^3$ . Use $q^3$ .

Compare $r^2$ and $r^3$ . Use $r^3$ .

  $\text{LCM}=78\cdot p^3\cdot q^3\cdot r^3=78p^3{q}^3{r}^3$

Therefore, the LCM of $26p^3{q}^2{r}^2,\text{ 39}{p}^2{q}^3{r}^2,$ and $\text{78}{p}^2{q}^2{r}^3$ is $78p^3{q}^3{r}^3$ .