Log In Start studying!

Select your language

Suggested languages for you:

Americas

English (US)

Europe

11

Expert-verified

Abstract Algebra: An Introduction

Found in: Page 14

Book edition 3rd

Author(s) Thomas W Hungerford, David Leep

Pages 608 pages

ISBN 9781111569624

Answers without the blur.

Just sign up for free and you're in.

Register for Free I’ll do it later

Short Answer

If $n \in ℤ$ ,what are the possible values of
(a) $(n, n + 2)$ (b) $(n, n + 6)$

(a) The possible values of $d$ are 1 and 2.

(b) The possible values of $d$ are 1, 2, 3, and 6.

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Definition

According to the definition, if $a, b \neq 0$ , then their greatest common divisor exists with a unique value. Thus, the greatest common divisor satisfies the condition $(a, b) \geq 1$ , since 1 is the divisor of both $a, b$ .

Solve for n,n+2

It is given that $(n, n + 6)$ .

Let $d = (n, n + 6)$ , then $d | 2$ and $d | n + 2$ , then there exist some integers $j$ and $k$ such that $j d = n$ and $k d = n + 2$ . Both the conditions imply that:

$\begin{array}{rcl} 2 & = & k d - j d \\ (n + 2) - n & = & d (k - j) \end{array}$

Hence, $d$ is a positive integer that divides 2; the values of $d$ can be 1 and 2, as both are factors of 2.

Solve for n,n+6

It is given that $(n, n + 6)$ .

Let role="math" localid="1646130808945" $d = (n, n + 6)$ . By following the similar procedure as in step 2, $d | 6$ , then the factors of 6 are 1, 2, 3, and 6 itself, then $d \in \{1, 2, 3, 6\}$ .

Hence, the possible values of are 1, 2, 3, and 6.

Most popular questions for Math Textbooks

Suppose that $a, b, c$ and $r$ are integers such that $a = b q + r$ . Prove each of the following statements.

(a) Every common divisor $c$ of a and b is also a common divisor of $b$ and $r$ .

(b) Every common divisor of $b$ and $r$ is also a common divisor of a and b.

(c) $(a, b) = (b, r)$

If $n$ is a positive integer, prove that there exist $n$ consecutive composite integers.

_{If $a > o$ and $b > 0$ , prove that $[a, b] = \frac{a b}{(a, b)}$ .}

Let a be a positive integer. If $\sqrt{a}$ is rational, prove that $\sqrt{a}$ is an integer.
Let r be a rational number and a an integer such that $r^{n} = a$ . Prove that r is an integer. [Part (a) is the case when $n = 2$ ].

Let a be any integer and let b and e be positive integers. Suppose that when a is divided by b, the quotient is q and the remainder is r, so that

$a = b q + r$ and $0 \leq r < b$ .

If ae is divided by bc, show that the quotient is q and the remainder is rc.

Want to see more solutions like these?

Sign up for free to discover our expert answers

Get Started - It’s free

Recommended explanations on Math Textbooks

94% of StudySmarter users get better grades.

Sign up for free