$x$ is odd if and only if $3x+6$ is odd 
Prove the following proposition. Let $x\in\Bbb Z$. Then $x$ is odd if and only if $3x+6$ is odd. 

I'm currently not seeing a way to transform $3x+6$ into the format of $2k+1$ in order to prove odd. This is my first time dealing with discrete mathematics and proofs and I'm trying to get a feel for it.
 A: Hint: Write $3x+6=x+2(x+3)$  .
A: Hints:
(1) Adding $2$ to an integer doesn't change its parity.
(2) Multiplying an integer by an odd number doesn't change its parity.
(3) You can write $3x+6=3(x+ 2)$
A: If $x$ is odd then it is of the form $2k+1$ for some integer $k$.   …   Thus $3x+6$ is of the form $2(\underline{\qquad})+1$ for some integer $(\underline{\qquad})$; meaning it is odd.
If $x$ is even then it is of the form $2k$ for some integer $k$.   …   Thus $3x+6$ is of the form $2(\underline{\qquad})$ for some integer $(\underline{\qquad})$; meaning that it is even.
Therefore $3x+6$ is odd if and only if $x$ is odd.
A: $3x+6$ is odd,we know $3x$ is odd ,so set $3x=2k+1$,$k=1,2,3...$,
so $$x=\frac{2k}{3}+1,k=1,2,3...$$
then  $\frac{2k}{3}$ is integer so $k=3n$,and $$x=2n+1,n=3k,k=1,2,3$$
and $x$ is  odd
A: Hints:
(1) An "if and only" statement can be split into two statements. You can prove "A if and only if B" by proving "if A then B" and "if B then A".
(2) A conditional statement is logically equivalent to its contrapositve. For example "if A then B" is logically equivalent to "if (not B) then (not A)".
(3) Using hints (1) and (2), one way to prove the statement you are asking about is to prove the following two statements:


*

*Let $x\in\Bbb Z$. If $x$ is odd, then $3x+6$ is odd.

*Let $x\in\Bbb Z$. If $x$ is even, then $3x+6$ is even.

A: Hint $ $ negate $\ 2\mid 3(x\!+\!2)\iff 2\mid x\!+\!2\iff 2\mid x$
A: We want to prove that $x$ is odd iff $3x+6$ is odd.  For many of these kinds of problems, the proof is easier if we start by assuming that $x$ is of the form $x=2k$ or $x=2k+1$ (these exhaust all possibilities for $x$) and substitute these expressions into $3x+6$.
Suppose $x$ is even, say $x=2k$ for some integer $k$. Then $3x+6 = 3(2k)+6=6x+6=2(3x+3)$, which is even.
Suppose $x$ is odd, say $x=2k+1$ for some integer $k$. Then $3x+6 = 3(2k+1)+6 = 6k+9 = 6k+8+1 = 2(3k+4)+1$, which is odd.
