1. ## Implication theorem

The following theorem has the form of an implication.

If n is an integer and 3n+2 is even, then n is even.

a) Prove this theorem by contradiction.
b) Give a direct proof of this theorem.
(Hint: it may be useful to get an equation for n in which
the term to which n is equal also includes n itself.)

2. Originally Posted by kashifzaidi
The following theorem has the form of an implication.

If n is an integer and 3n+2 is even, then n is even.

a) Prove this theorem by contradiction.
b) Give a direct proof of this theorem.
(Hint: it may be useful to get an equation for n in which
the term to which n is equal also includes n itself.)

Let $n\in\mathbb{Z}$ and let $3n+2$ be even.

Assume $n$ is not even. Therefore $n$ is odd. Therefore $n=2x+1$ for some $x\in\mathbb{Z}$

So $3n+2=3(2x+1)+2=6x+3+2=6x+5=2(3x+2)+1$

But $x$ is an integer, so $3x$ is an integer, so $3x+2$ is an integer. Thefore $2(3x+2)+1=2z+1$ for some $z\in\mathbb{Z}$

This is the definition of an odd number, so $3n+2$ is odd. But we know $3n+2$ is even. This is a contradiction and therefore the assumption that $n$ is not even is wrong. Therefore $n$ is even

Can you give the second part b try