Tübitak Lise 1. Aşama 2017 Soru 20

[Dogukan6336]

$ \sum _{n=1}^{30}n^{61}\equiv x \pmod {31^2} $ ise $x$ aşağıdakilerden hangisi olabilir? 

$\textbf{a)}\  404  \qquad\textbf{b)}\ 434  \qquad \textbf{c)}\ 465 \qquad \textbf{d)}\ 496  \qquad\textbf{e)}\ 527 $

[Metin Can Aydemir]

Cevap: $\boxed D$

$$(31-n)^{61}+n^{61}=31^{61}-31^{60}\cdot n\cdot \binom{61}{1}+\dots+n^{60}\cdot 31 \cdot \binom{61}{1} \equiv n^{60}\cdot 31\cdot 61(mod~31^2)$$ $$\sum\limits_{n=1}^{30} n^{61} \equiv 31\cdot 61\cdot \sum\limits_{n=1}^{15} n^{60}\equiv x\equiv 31a~(mod~31^2)  $$
$$61\cdot \sum\limits_{n=1}^{15} n^{60}\equiv -\sum\limits_{n=1}^{15} n^{60}\equiv -15 \equiv 16\equiv a~(mod~31) \Rightarrow x\equiv 31a\equiv 496~(mod~31^2)$$ bulunur.