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 #10
avatar+2234 
+5

Here’s the correct presentation. Previously, I transposed the number and modulus divisor.

An excess consumption of fermented bananas might be a major contributing factor in the cause.  I’ve left the previous presentation as a monument to my error (The math is fine). This is an extremely rare event: this is only the second time I’ve made a mistake.surprise  The first time was when I thought I made a mistake, but I didn’t, so I was wrong in thinking I did. smiley

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n231(mod3331)n1247(mod1361)Set m=33311361=4533491

 

n=2311361[1361φ(3331)1(mod3331)]=modulo inverse 1361 mod 3331=136133301mod3331=13613329mod3331=1980+12473331[3331φ(1361)1(mod1361)]=modulo inverse 3331 mod 1361=333113601mod1361=33311359mod1361=552n=2311361[1980]+12473331[552]n=622494180+2292873864n=2915368044n(modm)=2915368044(mod4533491)=333331n=333331+k4533491kZnmin=333331

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Nov 25, 2017
 #5
avatar+590 
+1
Nov 25, 2017
 #3
avatar+9488 
+2

a.  https://www.desmos.com/calculator/v50rjzsxtt

 

Since  a  can be any value > 0 , the graph will be different for each possible value of  a .

(You can pick a different value for  a  with the slider.)

 

b.  https://www.desmos.com/calculator/te6iwcz85v

 

Like the first graph, this graph will be different for each possible value of  b .

Nov 25, 2017
Nov 24, 2017
 #2
avatar+205 
+3
Nov 24, 2017
 #4
avatar+536 
0
Nov 24, 2017
 #4
avatar+118704 
0

 

Some interesting coding by GingerAle and Heureka:  Thanks to both of you :)

It is from this question:

https://web2.0calc.com/questions/modular-math_8

 

n=33311247[1247φ(231)1(mod231)]=modulo inverse 1247 mod 231=12472301mod231=1247229mod231=113+1361231[231φ(1247)1(mod1247)]=modulo inverse 231 mod 1247=23112461mod1247=2311245mod1247=637n=33311247[113]+1361231[637]n=469374541+200267067n=669641608n(modm)=669641608(mod288057)=197140n=197140+k288057kZnmin=197140

 

 

CODING:

 

\begin{array}{rcll} n &=& {\color{red}3331} \cdot {\color{green}1247} \cdot \underbrace{ \underbrace{ \underbrace{ \underbrace{ [ {\color{green}1247}^{\varphi({\color{green}231})-1} \pmod {{\color{green}231}} ] }_{=\text{modulo inverse 1247 mod 231} } }_{=1247^{230-1} \mod {231} }}_{=1247^{229} \mod {231}}}_{=113} + {\color{red}1361} \cdot {\color{green}231} \cdot \underbrace{ \underbrace{ \underbrace{ \underbrace{ [ {\color{green}231}^{\varphi({\color{green}1247})-1} \pmod {{\color{green}1247}} ] }_{=\text{modulo inverse 231 mod 1247} } }_{=231^{1246-1} \mod {1247} }}_{=231^{1245} \mod {1247}}}_{=637}\\\\

n &=& {\color{red}3331} \cdot {\color{green}1247} \cdot [ 113] + {\color{red}1361} \cdot {\color{green}231} \cdot [637] \\

n &=& 469374541 + 200267067 \\ n &=& 669641608 \\\\ && n\pmod {m}\\ &=& 669641608 \pmod {288057} \\

&=& 197140 \\\\

n &=& 197140 + k\cdot 288057 \qquad k \in Z\\\\

\mathbf{n_{min}} & \mathbf{=}& \mathbf{197140 }

\end{array}

Nov 24, 2017

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