Chapter 4.4, Problem 26WE

(a).

To determine

To find: the GCF of the given monomials.

Answer to Problem 26WE

The GCF of $200a^3{b}^{2}c,\text{ 300}{a}^{2}b{c}^3,$ and $\text{400}a{b}^3{c}^2$ is $100abc$ .

Explanation of Solution

Given information:

The given monomials:

  $200a^3{b}^{2}c,\text{ 300}{a}^{2}b{c}^3,\text{ 400}a{b}^3{c}^2$

Calculation:

The prime factorization of the coefficients are:

  $\begin{array}{l}200=2^3\cdot 5^2\\ 300=3\cdot 2^2\cdot 5^2\\ 400=2^4\cdot 5^2\end{array}$

The GCF of $200,\text{ 300,}$ and $400$ is

  $2^2{.5}^2=100$

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

Compare $a^3,a^2,$ and $a$ . Use $a$ .

Compare $b^2,b,$ and $b^3$ . Use $b$ .

Compare $c,c^3,$ and $c^2$ . Use $c$ .

  $\text{GCF}=100\cdot a\cdot b\cdot c=100abc$

Therefore, the GCF of $200a^3{b}^{2}c,\text{ 300}{a}^{2}b{c}^3,$ and $\text{400}a{b}^3{c}^2$ is $100abc$ .

(b).

To determine

The LCM of the given monomials.

Answer to Problem 26WE

The LCM of $200a^3{b}^{2}c,\text{ 300}{a}^{2}b{c}^3,$ and $\text{400}a{b}^3{c}^2$ is $1200a^3{b}^3{c}^3$ .

Explanation of Solution

Given information:

The given monomials:

  $200a^3{b}^{2}c,\text{ 300}{a}^{2}b{c}^3,\text{ 400}a{b}^3{c}^2$

Calculation:

As per problem,

The prime factorization of the coefficients are:

  $\begin{array}{l}200=2^3\cdot 5^2\\ 300=3\cdot 2^2\cdot 5^2\\ 400=2^4\cdot 5^2\end{array}$

The LCM of $200,\text{ 300,}$ and $400$ is

  $2^4\cdot 3\cdot 5^2=1200$

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

Compare $a^3,a^2,$ and $a$ . Use $a^3$ .

Compare $b^2,b,$ and $b^3$ . Use $b^3$ .

Compare $c,c^3,$ and $c^2$ . Use $c^3$ .

  $\text{LCM}=1200\cdot a^3\cdot b^3\cdot c^3=1200a^3{b}^3{c}^3$

Therefore, the LCM of $200a^3{b}^{2}c,\text{ 300}{a}^{2}b{c}^3,$ and $\text{400}a{b}^3{c}^2$ is $1200a^3{b}^3{c}^3$ .