Show that $2007^{2013}-1974^{2013}-1946^{2013}+1913^{2013}$ is divisible by 2013 
Let $$N = 2007^{2013}-1974^{2013}-1946^{2013}+1913^{2013}$$
Then select all the option(s) that are correct:

*

*N is divisible by 61

*N is divisible by 2013

*N is divisible by 28

*All of these


My attempt:
I tried to use the property $a^x - b^x = (a-b)(a^{x-1} + ... + b^{x-1})$ for odd x. Note that $2007 - 1946 = 61$ and $1974-1913 = 61$
$$N = 61(2007^{2012} +\ ...\ + 1946^{2012} - (1974^{2012} +\ ...\ + 1913^{2012})) $$
Option 1 is correct. However, the answer key says that $N$ is also divisible by $2013$. How do I prove this? Little fermat won't work because $2013$ is not prime.
 A: We have $$2013=61\cdot33.$$
Since $$2007-1946=1974-1913=61$$
and $$N=2007^{2013}-1946^{2013}-\left(1974^{2013}-1913^{2013}\right),$$ we see that $N$ is divisible by $61$.
Thus, it's enough to prove that $N$ is divisible by $33.$
Now, write $$N=2007^{2013}-1974^{2013}-\left(1946^{2013}-1913^{2013}\right).$$
Can you end it now?
Also, since $$N=2007^{2013}+1913^{2013}-\left(1946^{2013}+1974^{2013}\right),$$
we see that $N$ is divisible by $28$.
A: Corect is "All" by $\,m,n,d = 61,33,1913\,$ below, $ $  & $\bmod 28\!:\ m\!+\!n\!+\!2d\equiv 5\!+\!5\!+\!2\cdot 9\equiv 0$
$$\begin{align}
&\ \ \ \ \ \ \ \ \ \ 2007^k  -\ \ \ \ \ 1974^k -\ \ \ 1946^k +1913^k\\[.1em]
=\ \  f =\         &(\color{#c00}m\!+\!n\!+\!d)^k-(\color{#c00}m\!+\!d)^k-(n\!+\!d)^k+d^k\\[.2em]
\Longrightarrow\ f\,\equiv\ &\ \ \ \ \ \ \  (n\!+\!d)^k \ \ \ \ -\ \ \ \ \  d^k\: - (n\!+\!d)^k + d^k\equiv 0\pmod{\color{#c00}m}\\[.1em]
\&\ \ \ f\,\equiv\ &\ \ \ \ \ \ (m\!+\!d)^k - (m\!+\!d)^k  \ \ \ \ - \ \ \  d^k\ +\ d^k\equiv 0\pmod{n}\\[.1em]
\&\ \ \ f\,\equiv\ &\ \ \ \ \ \ \ \ \ \ (-d)^k\!\! +\! (-n\!-\!d)^k\! + (n\!+\!d)^k +\, d^k\equiv 0\pmod{m\!+\!n\!+\!2d},\  {\rm by}\ \ k\ \rm odd
\end{align}\qquad$$
Remark $ $ Thus for any integers $\,m,n,d\,$ the above $\,f(m,n,d)\,$is divisible by $m,n$, and also by $\,m\!+\!n\!+\!2d$ if $k$ is odd. See this answer for more on the innate symmetry at the heart of the matter.
