Chapter 4.4, Problem 31WE

a.

To determine

To state:the positive factors of the given product.

Answer to Problem 31WE

The factors of pqare $1$ , p, q, and pq.

Explanation of Solution

Given:

pand q represent different prime numbers.

The given product:

pq

Calculation:

As per the given problem

The given product pq has two prime factors p and q.

The product can be factored in the following ways:

( $1$ , pq) and (p,q)

Thus, the factors of pqare $1$ , p, q, and pq.

b.

To determine

To state:the positive factors of the given product.

Answer to Problem 31WE

The factors of $p^2{q}^2$ are $1$ , p, q, $p^2,q^2$ , pq, $p^{2}q,p{q}^2,$ and $p^2{q}^2$ .

Explanation of Solution

Given:

pand q represent different prime numbers.

The given product:

  $p^2{q}^2$

Calculation:

As per the given problem

The given product $p^2{q}^2$ has two prime factors p and q.

The productcan be factored in the following ways:

  $(1,p^2{q}^2),\text{ (}p,p{q}^2),\text{ (}q,p^{2}q),(pq,pq),\text{}\text{}\text{ and}\text{}\text{ (}{p}^2,q^2)$

Thus, the factors of $p^2{q}^2$ are $1$ , p, q, $p^2,q^2$ , pq, $p^{2}q,p{q}^2,$ and $p^2{q}^2$ .

c.

To determine

To state:the positive factors of the given product.

Answer to Problem 31WE

The factors of $p^3{q}^3$ are $1$ , p, q, $p^2,q^2$ , pq, $p^3,q^3,p^{2}q,p{q}^2,p{q}^3,p^{3}q,p^2{q}^2,p^2{q}^3,p^3{q}^2,$ and $p^3{q}^3$ .

Explanation of Solution

Given:

pand q represent different prime numbers.

The given product:

  $p^3{q}^3$

Calculation:

As per the given problem

The given product $p^3{q}^3$ has two prime factors p and q.

The product can be factored in the following ways:

  $(1,p^3{q}^3),\text{ (}p,p^2{q}^3),\text{ (}q,p^3{q}^2),(p^2,p{q}^3),\text{ (}{q}^2,p^{3}q),(pq,p^2{q}^2),(p^{2}q,p{q}^2),\text{ and }\text{}(p^3,q^3)\text{}$

Thus, the factors of $p^3{q}^3$ are $1$ , p, q, $p^2,q^2$ , pq, $p^3,q^3,p^{2}q,p{q}^2,p{q}^3,p^{3}q,p^2{q}^2,p^2{q}^3,p^3{q}^2,$ and $p^3{q}^3$ .