Chứng minh: Số A = 12^2012 – 2^2000 chia hết cho 10? 24/08/2021 Bởi Lyla Chứng minh: Số A = 12^2012 – 2^2000 chia hết cho 10?
Giải thích các bước giải: \[\begin{array}{l} + )12 \equiv 2\left( {\bmod 10} \right)\\ \Rightarrow {12^{2012}} \equiv {2^{2012}}\left( {\bmod 10} \right)\\ \Rightarrow {12^{2012}} – {2^{2000}} \equiv {2^{2012}} – {2^{2000}}\left( {\bmod 10} \right)\\ + ){2^{2012}} – {2^{2000}} = {2^{2000}}\left( {{2^{12}} – 1} \right)\\{2^4} \equiv 1\left( {\bmod 5} \right) \Rightarrow {2^{12}} = {2^{4.3}} = {\left( {{2^4}} \right)^3} \equiv 1\left( {\bmod 5} \right)\\ \Rightarrow {2^{12}} – 1 \equiv 0\left( {\bmod 5} \right)\\ \Rightarrow {2^{2000}}\left( {{2^{12}} – 1} \right) \equiv 0\left( {\bmod 10} \right)\\ \Rightarrow {2^{2012}} – {2^{2000}} \equiv 0\left( {\bmod 10} \right)\\ \Rightarrow {12^{2012}} – {2^{2000}} \equiv 0\left( {\bmod 10} \right)\\ \Rightarrow {12^{2012}} – {2^{2000}} \vdots 10\end{array}\] Bình luận
Giải thích các bước giải:
\[\begin{array}{l}
+ )12 \equiv 2\left( {\bmod 10} \right)\\
\Rightarrow {12^{2012}} \equiv {2^{2012}}\left( {\bmod 10} \right)\\
\Rightarrow {12^{2012}} – {2^{2000}} \equiv {2^{2012}} – {2^{2000}}\left( {\bmod 10} \right)\\
+ ){2^{2012}} – {2^{2000}} = {2^{2000}}\left( {{2^{12}} – 1} \right)\\
{2^4} \equiv 1\left( {\bmod 5} \right) \Rightarrow {2^{12}} = {2^{4.3}} = {\left( {{2^4}} \right)^3} \equiv 1\left( {\bmod 5} \right)\\
\Rightarrow {2^{12}} – 1 \equiv 0\left( {\bmod 5} \right)\\
\Rightarrow {2^{2000}}\left( {{2^{12}} – 1} \right) \equiv 0\left( {\bmod 10} \right)\\
\Rightarrow {2^{2012}} – {2^{2000}} \equiv 0\left( {\bmod 10} \right)\\
\Rightarrow {12^{2012}} – {2^{2000}} \equiv 0\left( {\bmod 10} \right)\\
\Rightarrow {12^{2012}} – {2^{2000}} \vdots 10
\end{array}\]