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 #4
avatar+2489 
+9

Solution for (n) (smallest positive integer that satisfies the system of congruencies).

n mod 1103 = 1041

n mod 1303 = 859

n mod 2003 = 1095

 

\(\begin{array}{rcll} n &\equiv& {\color{red}1041} \pmod {{\color{green}1103}} \\ n &\equiv& {\color{red}859} \pmod {{\color{green}1303}} \\n &\equiv& {\color{red}1095} \pmod {{\color{green}2003}} \\ \text{Let } m &=&1103 \cdot 1303\cdot 2003 = 2878729627 \\ \end{array} \)

\(\text {1103, 1303, and 2003 are coprime numbers (they are actually prime).}\\\)

\(\small{ \begin{array}{l} n = {\color{red}1041} \cdot {\color{green}1303\cdot 2003} \cdot \underbrace{ \underbrace{ \underbrace{ \underbrace{ \underbrace{ \underbrace{ [ { (\color{green}1303\cdot 2003)}^{\varphi({\color{green}1103}) -1 } \pmod {{\color{green}1103}} ] }_{=\text{modulo inverse }(1303\cdot 2003) mod 1103 }}_{=(1303\cdot 2003)^{1103-1} \mod {1103}} }_{=(1303\cdot 2003)^{1102} \mod {1103}} }_{=(2609909\pmod{1103})^{1102} \mod {1103}} }_{=(211)^{1102} \mod {1103}} }_{=988} + {\color{red}859} \cdot {\color{green}1103\cdot 2003} \cdot \underbrace{ \underbrace{ \underbrace{ \underbrace{ \underbrace{ \underbrace{ [ { (\color{green}1103\cdot 2003) }^{\varphi({\color{green}1303}) -1} \pmod {{\color{green}1303}} ] }_{=\text{modulo inverse } (1103\cdot 2003) mod 1303 } }_{=(1103\cdot 2003)^{1302-1} \mod {1303}} }_{=(1103\cdot 2003)^{1301} \mod {1303}} }_{=(2209309\pmod{1303})^{1301} \mod {1303}} }_{=(724)^{1301} \mod {1303}} }_{=9} +{\color{red}{1095}} \cdot {\color{green}1103\cdot 1303} \cdot \underbrace{ \underbrace{ \underbrace{ \underbrace{ \underbrace{ \underbrace{ [ { (\color{green}1103\cdot 1303) }^{\varphi({\color{green}2003}) -1 } \pmod {{\color{green}2003}} ] }_{=\text{modulo inverse } (1103\cdot 1303) mod 2003 } }_{=(1103\cdot 1303)^{2002-1} \mod {2003}} }_{=(1103\cdot 1303)^{2001} \mod {2003}} }_{=(1437209\pmod{2003})^{2001} \mod {2003}} }_{=(1058)^{2001} \mod {2003}} }_{=195}\\\\ n = {\color{red}{1041}} \cdot {\color{green}{1303}\cdot 2003} \cdot [988] + {\color{red}859} \cdot {\color{green}1103\cdot 2003} \cdot [9] + {\color{red}1095} \cdot {\color{green}1103\cdot 1303} \cdot [195] \\ n = 2684312285772 + 17080167879 + 306880051725 \\ n = 3008272505376 \\\\ n \pmod {m}\\ = 3008272505376 \pmod {2878729627} \\ = 45161 \\\\ n = 45161 + k\cdot 2878729627 \qquad k \in Z\\\\ \mathbf{n_{min}} \mathbf{=} \mathbf{45161} \end{array}} \)

 

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\(\small \text {Related formulas and principles compliments of Leonhard Euler }\scriptsize \text {(totient function),} \\ \small \text {Euclid of Alexandria }\scriptsize \text {(Extended Euclidean algorithm), and Brilliant Chinese mathematicians “Chinese Remainder Theorem” } \\ \small \text {LaTex layout and coding adapted from Heureak’s mathematical solution and Latex presentation:}\\ \tiny \text {http://web2.0calc.com/questions/find-the-smallest-positive-integer-that-satisfies-the-system-of-congruences }\\ \small \text {Produced by Lancelot Link & Co.}\\ \small \text {Directed by GingerAle}\\ \small \text {Sponsored by Nause Corp pharmaceuticals: Makers of } \\ \scriptsize \text { Quantum Vaccines for spooky dumbness at a distance and related contagious dumbness diseases (rCDDs).}\\ \scriptsize \text{ and}\\ \scriptsize \text{ Master Blarney Filters. Now filters most toxins emitted by blarney bankers and related dumb-dumbs. } \\ \)

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Mar 13, 2017
 #5
avatar+118658 
+10

Hi again Sara, thank you for giving me what I asked for.  I can see that you did ask for more help before too, just like you said :)

It is very easy for things like this to be overlooked so it was good that you reposted.  :)

 

You have lots of answers now but if it is still not clear to you then say so.  :))

 

If you would like me, or someone else to look at it specifically for you, you can also message us privately, don't forget to include the address as well though so we can find it :)

If you have specific questions about someone's answer then aks them on the forum and maybe back it up with a private message :)

Mar 13, 2017
 #5
avatar+9477 
+11

Ohh... thank you smiley

Mar 13, 2017

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