Sum of $k$ consecutive integers I'm trying to prove or disprove:
Characterize $k \in\Bbb N$ that makes the following statement true: “$a$ is
divisible by $k$ if and only if $a$ can be written as a sum of $k$ consecutive integers”.
I tried the converse:

Proof. If $k\mid a$ and $k=2q+1$, $q \in\Bbb Z $, then $a=c(2q+1)$ for $c \in\Bbb Z$.
Notice: $$\begin{align*}ck &= (c - q) + \big(c - (q-1)\big) + \ldots+ (c -1) + c\\&+\, (c +1) +\ldots+ \big(c + (q-1)\big) + (c + q)\;,\end{align*}$$ so $ck$ is the sum of the $2q+1$ consecutive integers.
Hence $a$ is expressable as a sum of $k$ consecutive integers

Is this okay? Other methods?
 A: Since the statement you are trying to prove contains "if and only if", you also need to prove 
"if $a$ can be written as the sum of $k$ consecutive integers, $a$ is divisible by $k$."
Also, I'm not sure if it's just me being tired, but I'm not following your 'Notice:' line of reasoning.
A: You were asked to find a characterization of those positive integers $k$ that have a certain property; this means that you’re to find and prove a statement of the form

a positive integer $k$ has the specified property if and only if $k$ is something.

What you’ve proved is that $k$ has the desired property if $k$ is odd; to complete the characterization you must show that if $k$ is even, it does not have the specified property. That means that you must show that if $k$ is an even positive integer, then there is some integer $a$ such that either 


*

*$k\mid a$, but $a$ cannot be written as the sum of $k$ consecutive integers, or  

*$k\nmid a$, but $a$ can be written as the sum of $k$ consecutive integers.


HINT: What if $a=k$?
