Question

Tìm số dư chia 4362⁴³⁶² cho 11.

Answer 1

43622≡3(mod11)⇒43624362=43622.2181=32181" role="presentation" style="border:0px; color:rgb(40, 40, 40); direction:ltr; display:inline-block; float:none; font-family:helvetica,arial,sans-serif; font-size:16.38px; line-height:0; margin:0px; max-height:none; max-width:none; min-height:0px; min-width:0px; padding:1px 0px; position:relative; white-space:nowrap; word-wrap:normal" class="MathJax_CHTML mjx-chtml">43622≡3(mod11)⇒43624362=43622.2181=32181
35≡1(mod11)⇒32181=35.436+1=35.436.3≡1.3(mod11)=3" role="presentation" style="border:0px; color:rgb(40, 40, 40); direction:ltr; display:inline-block; float:none; font-family:helvetica,arial,sans-serif; font-size:16.38px; line-height:0; margin:0px; max-height:none; max-width:none; min-height:0px; min-width:0px; padding:1px 0px; position:relative; white-space:nowrap; word-wrap:normal" class="MathJax_CHTML mjx-chtml">35≡1(mod11)⇒32181=35.436+1=35.436.3≡1.3(mod11)=3
43624362≡3(mod11)" role="presentation" style="border:0px; color:rgb(40, 40, 40); direction:ltr; display:inline-table; float:none; font-family:helvetica,arial,sans-serif; font-size:16.38px; line-height:0; margin:0px; max-height:none; max-width:none; min-height:0px; min-width:0px; padding:1px 0px; position:relative; white-space:nowrap; word-wrap:normal" class="MathJax_CHTML mjx-chtml">43624362≡3(mod11) 

Answer 2

Ta có :

\(4362^{4362}=\left(6.727\right)^{4362}\)

\(=6^{4362}.727^{4362}\)

\(=\left(6^{10}\right)^{436}.6^2.\left(727^2\right)^{2181}\)

Ta có :

\(6^{10}\text{≡}1\left(mod11\right)\)

\(\Rightarrow\left(6^{10}\right)^{436}\text{≡}1\left(mod11\right)\)

\(727^2\text{≡}1\left(mod11\right)\)

\(\Rightarrow\left(727^2\right)^{2181}\text{≡}1\left(mod11\right)\)

\(\Rightarrow4362^{4362}\text{≡}1.6^2.1\text{≡}3\left(mod11\right)\)

Vậy ...

 

 

Answer 3

Ta có : 
4362² = 3 ( mod11 ) => 4362⁴³⁶² = 4362^2.2181 = 3²¹⁸¹ = 6⁴³⁶² . 72
Lại có : = ( 6¹⁰ ) ⁴³⁶ . 6² . ( 727² ) ²¹⁸¹
Do đó 4362⁴³⁶² = 3 (mod11) 6¹⁰ = 1 (mod11)