Chapter 1.5, Problem 14E

To determine

(a)

To write:

A counter example for below definition and define to all odd integers are sane.

An integer $n$ is sane if $3|(n^2+2n)$.

To determine

(b)

To show:

A direct proof of below definition for if $3|n$ then $n$ is sane:

An integer $n$ is sane if $3|(n^2+2n)$.

To determine

(c)

To show:

The proof by contradiction, if $n=3j+2$ for same integer $j$ then $n$ is sane by using below definition:

An integer $n$ is sane if $3|(n^2+2n)$.