Tübitak Lise 1. Aşama 2013 Soru 18

[alpercay]

$$\binom{2013}{1}+2013 \binom{2013}{3}+2013^2\binom{2013}{5}+\dots +2013^{1006}\binom{2013}{2013}$$ toplamının $41$ ile bölümünden kalan kaçtır?

$\textbf{a)}\ 20
\qquad\textbf{b)}\ 14
\qquad\textbf{c)}\ 7
\qquad\textbf{d)}\ 1
\qquad\textbf{e)}\ \text{Hiçbiri}
$

[geo]

Sorudaki toplama $A$ diyelim.
$$ \begin{array}{rcl}
(2+1)^{2013} + (2-1)^{2013} &=& 2\cdot {2013 \choose 0} \cdot 2^{2013} + 2\cdot {2013 \choose 2} \cdot 2^{2011} + \dots + 2\cdot {2013 \choose 2012} \cdot 2^{1} \\ \\
&=& {2013 \choose 2013} \cdot 4^{1007} + {2013 \choose 2011} \cdot 4^{1006} + \dots + {2013 \choose 1} \cdot 4^{1} \\ \\
&\equiv& 4A \pmod {41}
\end{array}$$
$4A \equiv 3^{2013} + 1  \equiv (3^4)^{503}\cdot 3 + 1 \equiv (-1)\cdot 3 + 1 \equiv -2 \equiv 80 \pmod {41}$

$\Rightarrow A \equiv 20 \pmod {41}$.