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What is the smallest positive integer n such that \(3n \equiv 1356 \pmod{22}? \)

 Jul 13, 2020
 #1
avatar+744 
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 only try to post a few or a single questions at a time, please

 Jul 13, 2020
 #3
avatar+279 
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yeah forumofweb2.0cal do your own homework

madyl  Jul 13, 2020
 #4
avatar+2313 
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I think this is Alcumus....

CalTheGreat  Jul 13, 2020
 #2
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I think it is 12.

 Jul 13, 2020
 #5
avatar+25543 
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What is the smallest positive integer n such that  \(3n \equiv 1356 \pmod{22}\)

 

\(\begin{array}{|rcll|} \hline 3n &\equiv& 1356 \pmod{22} \\ 3n &\equiv& 1356-61*22 \pmod{22} \\ 3n &\equiv& 14 \pmod{22} \\ n &\equiv& \dfrac{14}{3} \pmod{22} \\ n &\equiv& 14*3^{-1} \pmod{22} \\ && \boxed{ 3^{-1} \pmod{22} = 3^{\phi(22)-1} \pmod{22} \quad | \quad \phi(22)=22*\left( 1-\dfrac{1}{2}\right)*\left( 1-\dfrac{1}{11}\right)=10 \\ =3^{10-1} \pmod{22} \\ = 3^9 \pmod{22} \\ = 19683 \pmod{22} \\ = 19683-894*22 \pmod{22} \\ = 15 \pmod{22} \\\mathbf{3^{-1} \pmod{22}=15 \pmod{22}} } \\ n &\equiv& 14*15 \pmod{22} \\ n &\equiv& 210 \pmod{22} \\ n &\equiv& 210-9*22 \pmod{22} \\ \mathbf{n} &\equiv& \mathbf{12 \pmod{22}} \\ \hline \end{array} \)

 

The smallest positive integer n is \(\mathbf{12}\)

 

laugh

 Jul 14, 2020
edited by heureka  Jul 14, 2020

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