Regarding properties of function Dedekind sum in analytic number theory I am self studying Tom M. Apostol's Modular Functions and Dirichlet Series in Number Theory and I could not think about a theorem given in Chapter 3. Its proof has been omitted by Apostol. 
The theorem is:

If $hh' \equiv +1 \pmod{k}$, then $s(h', k) = + s(h, k)$ and if $hh'\equiv -1 \pmod{k}$, then $s(h', k) = -s(h, k)$. 

Here $s(h, k)$ is called Dedekind sum which is equal to
$$s(h,k) = \sum_{r=0}^k (( r/k)) ((hr/k))$$ where $(( x)) = x -[x] - 0.5$ if $x$ is not an integer and $((x)) = 0$ if $x$ is an integer, $[x]$ is the greatest integer less than or equal to x. 
Please give some hint. 
 A: Note that the function $((\,\cdot\,))$ is odd and has period $1$. Also, since $((x)) = 0$ when$x$ is an integer, the summands for $r = 0$ and $r = k$ vanish, thus we can write $s(h,k)$ in the more conventional form(s)
$$s(h,k) = \sum_{r = 0}^{k-1} \biggl(\biggl(\frac{r}{k}\biggr)\biggr)\biggl(\biggl(\frac{hr}{k}\biggr)\biggr) = \sum_{r = 1}^k \biggl(\biggl(\frac{r}{k}\biggr)\biggr)\biggl(\biggl(\frac{hr}{k}\biggr)\biggr)$$
and have the equality
$$s(h,k) = \sum_{r = 1}^{k-1} \biggl(\biggl(\frac{r}{k}\biggr)\biggr)\biggl(\biggl(\frac{hr}{k}\biggr)\biggr)\,.$$
By the periodicity we also have
$$s(h,k) = \sum_{r \bmod k} \biggl(\biggl(\frac{r}{k}\biggr)\biggr)\biggl(\biggl(\frac{hr}{k}\biggr)\biggr)$$
where the sum extends over an arbitrary complete residue system modulo $k$ (and the class $r \equiv 0 \pmod{k}$ may be included or omitted as convenience suggests).
With these preliminary remarks, in particular the last, we see that for every $a$ coprime to $k$ we may also write
$$s(h,k) = \sum_{t = 0}^{k-1} \biggl(\biggl(\frac{at}{k}\biggr)\biggr)\biggl(\biggl(\frac{hat}{k}\biggr)\biggr)$$
because $at$ runs through a complete residue system modulo $k$ if and only if $t$ does.
In this representation, take $a = h'$ (by the assumption $hh' \equiv \pm 1 \pmod{k}$ it follows that $h'$ is coprime to $k$) and reduce $hh't$ in the second factor of each summand modulo $k$. Thus
\begin{align}
s(h,k) &= \sum_{t = 0}^{k-1} \biggl(\biggl(\frac{h't}{k}\biggr)\biggr)\biggl(\biggl(\frac{hh't}{k}\biggr)\biggr) \\
&= \sum_{t = 0}^{k-1} \biggl(\biggl(\frac{h't}{k}\biggr)\biggr)\biggl(\biggl(\frac{\pm t}{k}\biggr)\biggr) \\
&= \sum_{t = 0}^{k-1} \biggl(\biggl(\frac{\pm t}{k}\biggr)\biggr)\biggl(\biggl(\frac{h't}{k}\biggr)\biggr) \\
&= \pm\sum_{t = 0}^{k-1} \biggl(\biggl(\frac{t}{k}\biggr)\biggr)\biggl(\biggl(\frac{h't}{k}\biggr)\biggr) \\
&= \pm s(h',t)\,,
\end{align}
where the sign is $+$ or $-$ according to whether $hh' \equiv 1 \pmod{k}$ or $hh' \equiv -1\pmod{k}$, and in the latter case the oddness of $((\,\cdot\,))$ was used at the penultimate equals sign.
