4. Let $n$ be a positive integer. Find the number of possible remainders when
$$
2013^{n}-1803^{n}-1781^{n}+1774^{n}
$$
is divided by 203 .
(A) 1 ;
(B) 2 ;
(C) 3 ;
(D) 4 ;
(E) More than 4.
4. Answer: (A).
For any positive integer $n$,
$$
\begin{aligned}
2013^{n}-1803^{n}-1781^{n}+1774^{n} & =\left(2013^{n}-1803^{n}\right)-\left(1781^{n}-1774^{n}\right) \\
& =(2013-1803) u-(1781-1774) v=210 u-7 v, \\
2013^{n}-1803^{n}-1781^{n}+1774^{n} & =\left(2013^{n}-1781^{n}\right)-\left(1803^{n}-1774^{n}\right) \\
& =(2013-1781) x-(1803-1774) y=29 x-29 y .
\end{aligned}
$$

So $2013^{n}-1803^{n}-1781^{n}+1774^{n}$ is divisisle by $7 \times 29=203$ for every posit ive integer $n$.