Solve ##x\equiv 5\pmod{11},x\equiv 14\pmod{29},x\equiv 15\pmod{31}##

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The simultaneous congruences x≡5 (mod 11), x≡14 (mod 29), and x≡15 (mod 31) are solved using the Chinese Remainder Theorem, resulting in n=9889. The values N_k are calculated as N_1=899, N_2=341, and N_3=319. The modular inverses are found to be x_1=7, x_2=4, and x_3=7. The final solution is x≡4944 (mod 9889), confirmed through calculations and checks using modulo operations. The method effectively demonstrates the application of the Chinese Remainder Theorem for solving simultaneous congruences.
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Homework Statement
Solve the following set of simultaneous congruences:
##x\equiv 5\pmod{11},x\equiv 14\pmod {29},x\equiv 15\pmod{31}##.
Relevant Equations
None.
Consider the following set of simultaneous congruences:
## x\equiv 5\pmod {11}, x\equiv 14\pmod {29}, x\equiv 15\pmod {31} ##.
Applying the Chinese Remainder Theorem produces:
## n=11\cdot 29\cdot 31=9889 ##.
Now we define ## N_{k}=\frac{n}{n_{k}} ## for ## k=1, 2,..., r ##.
Observe that ## N_{1}=\frac{9889}{11}=899, N_{2}=\frac{9889}{29}=341 ## and ## N_{3}=\frac{9889}{31}=319 ##.
Then
\begin{align*}
&899x_{1}\equiv 1\pmod {11}\\
&341x_{2}\equiv 1\pmod {29}\\
&319x_{3}\equiv 1\pmod {31}.\\
\end{align*}
This means ## x_{1}=7, x_{2}=4 ## and ## x_{3}=7 ##.
Thus ## x\equiv (5\cdot 899\cdot 7+14\cdot 341\cdot 4+15\cdot 319\cdot 7)\pmod {9889}\equiv 84056\pmod {9889}\equiv 4944\pmod {9889} ##.
Therefore, ## x\equiv 4944\pmod {9889} ##.
 
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Math100 said:
Homework Statement:: Solve the following set of simultaneous congruences:
## x\equiv 5\pmod {11}, x\equiv 14\pmod {29}, x\equiv 15\pmod {31} ##.
Relevant Equations:: None.

Consider the following set of simultaneous congruences:
## x\equiv 5\pmod {11}, x\equiv 14\pmod {29}, x\equiv 15\pmod {31} ##.
Applying the Chinese Remainder Theorem produces:
## n=11\cdot 29\cdot 31=9889 ##.
Now we define ## N_{k}=\frac{n}{n_{k}} ## for ## k=1, 2,..., r ##.
Observe that ## N_{1}=\frac{9889}{11}=899, N_{2}=\frac{9889}{29}=341 ## and ## N_{3}=\frac{9889}{31}=319 ##.
Then
\begin{align*}
&899x_{1}\equiv 1\pmod {11}\\
&341x_{2}\equiv 1\pmod {29}\\
&319x_{3}\equiv 1\pmod {31}.\\
\end{align*}
This means ## x_{1}=7, x_{2}=4 ## and ## x_{3}=7 ##.
Thus ## x\equiv (5\cdot 899\cdot 7+14\cdot 341\cdot 4+15\cdot 319\cdot 7)\pmod {9889}\equiv 84056\pmod {9889}\equiv 4944\pmod {9889} ##.
Therefore, ## x\equiv 4944\pmod {9889} ##.
Correct, and I checked all calculations. Btw., how did you find ##x_1,x_2## and ##x_3##?

Edit: A way to check modulo operations with calc.exe from windows is e.g. ##4944 \, : \,31= 159,...## followed by ##159,... \, - \,159 = ...## and ##... \, \cdot \,31 =15.## Hence ##4944 \equiv 15\pmod{31}.##
 
fresh_42 said:
Correct, and I checked all calculations. Btw., how did you find ##x_1,x_2## and ##x_3##?

Edit: A way to check modulo operations with calc.exe from windows is e.g. ##4944 \, : \,31= 159,...## followed by ##159,... \, - \,159 = ...## and ##... \, \cdot \,31 =15.## Hence ##4944 \equiv 15\pmod{31}.##

\begin{align*}
&899x_{1}\equiv 1\pmod {11}\\
&\implies 8x_{1}\equiv 1\pmod {11}\\
&\implies 32x_{1}\equiv 4\pmod {11}\\
&\implies -x_{1}\equiv 4\pmod {11}\\
&\implies x_{1}\equiv -4\pmod {11}\\
&\implies x_{1}\equiv 7\pmod {11}.\\
\end{align*}

## x_{1}=7 ##

\begin{align*}
&341x_{2}\equiv 1\pmod {29}\\
&\implies 22x_{2}\equiv 1\pmod {29}\\
&\implies -7x_{2}\equiv 1\pmod {29}\\
&\implies -28x_{2}\equiv 4\pmod {29}\\
&\implies x_{2}\equiv 4\pmod {29}.\\
\end{align*}
## x_{2}=4 ##
\begin{align*}
&319x_{3}\equiv 1\pmod {31}\\
&\implies 9x_{3}\equiv 1\pmod {31}\\
&\implies 36x_{3}\equiv 4\pmod {31}\\
&\implies 5x_{3}\equiv 4\pmod {31}\\
&\implies -30x_{3}\equiv -24\pmod {31}\\
&\implies x_{3}\equiv 7\pmod {31}.\\
\end{align*}
## x_{3}=7 ##
 
Last edited by a moderator:
Another way. Let
x=31a+15=29a+(2a+15)=11*3a+(-2a+15)
So
2a \equiv -1(mod\ 29)
2a \equiv -1(mod\ 11)
The least a is
2a+1=29*11
a=159
The least x is
x=31*159+15=4944
In general
x=31*29*11b+4944
where b=0,1,2,...
 
Last edited:
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