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Chứng minh rằng 32010+52010chia hết cho 13

 

shitbo
19 tháng 6 2019 lúc 10:18

\(3^3=27\equiv1\left(mod13\right)\Rightarrow\left(3^3\right)^{670}\equiv1^{670}\equiv1\left(mod13\right)\) 

\(\equiv5^2=25\equiv-1\left(mod13\right)\Rightarrow\left(5^2\right)^{1005}\equiv\left(-1\right)^{1005}\left(mod13\right)\) 

\(\Rightarrow3^{2010}+5^{2010}\equiv\left(-1\right)+1\equiv0\left(mod13\right)\Rightarrowđpcm\)

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52010=(52)1005≡(−1)1005(mod13)" role="presentation" style="border:0px; color:rgb(40, 40, 40); direction:ltr; display:inline-table; float:none; font-family:helvea,arial,sans-serif; font-size:16.38px; line-height:0; margin:0px; max-height:none; max-width:none; min-height:0px; min-width:0px; overflow-wrap:normal; padding:1px 0px; position:relative; white-space:nowrap" class="MathJax_CHTML mjx-chtml">
32010+52010" role="presentation" style="border:0px; color:rgb(40, 40, 40); direction:ltr; display:inline-block; float:none; font-family:helvea,arial,sans-serif; font-size:16.38px; line-height:0; margin:0px; max-height:none; max-width:none; min-height:0px; min-width:0px; overflow-wrap:normal; padding:1px 0px; position:relative; white-space:nowrap" class="MathJax_CHTML mjx-chtml"> chia hết cho 13

Ta có \(3^{2010}=\left(3^3\right)^{670}=1^{670}\left(mod13\right)\)

Mà \(5^{2010}=\left(5^2\right)^{1005}=\left(-1\right)^{1005}\left(mod13\right)\)

\(\Rightarrow3^{2010}+5^{2010}⋮13\)