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### Abstract Algebra: An Introduction

Book edition 3rd
Author(s) Thomas W Hungerford, David Leep
Pages 608 pages
ISBN 9781111569624

# If $n\in \mathrm{ℤ}$,what are the possible values of (a) $\left(n,n+2\right)$ (b) $\left(n,n+6\right)$

(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

## Definition

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

## Solve for n,n+2

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

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

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

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 $\left(n,n+6\right)$ .

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

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