We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive pseudonymised information about your use of our website. cookie policy and privacy policy.
 
+0  
 
0
204
1
avatar

Find \(24^{-1} \pmod{11^2}\). That is, find the residue b`$ for which \(24b \equiv 1\pmod{11^2}\). Express your answer as an integer from 0 to \(11^2-1\), inclusive.

I would really appreciate if this is done fast!!

 Aug 27, 2018

Best Answer 

 #1
avatar+22188 
+4

Modulo inverses
Find
\(24^{-1} \pmod{11^2}\).
That is, find the residue b  for which \(24b \equiv 1\pmod{11^2}\).
Express your answer as an integer from 0 to , inclusive.

 

\(\begin{array}{|rcll|} \hline & 24\cdot \underbrace{24^{-1}}_{=b} \equiv 1 \pmod{11^2} \\ \hline \end{array}\)

 

\(\begin{array}{rcll} \text{Let} \\ & 24\cdot 5 = 120 \equiv -1 \pmod{11^2} \\ \end{array} \)

 

\(\begin{array}{llcll} \text{square this equation: } \\ & (24\cdot 5)\cdot(24\cdot 5) &\equiv& (-1)\cdot(-1) \pmod{11^2} \\ & 24\cdot (5^2\cdot 24) &\equiv& 1 \pmod{11^2} \\ & 24\cdot 600 &\equiv& 1 \pmod{11^2} \quad | \quad 600 \equiv 116 \pmod{11^2} \\ & 24\cdot \underbrace{116}_{=(24)^{-1}=b} &\equiv& 1 \pmod{11^2} \\ \end{array} \)

 

\(\text{So $24^{-1} \pmod{11^2}=\boxed{116}$ is the multiplicative inverse to $24$ modulo $11^2$}.\)

 

laugh

 Aug 28, 2018
 #1
avatar+22188 
+4
Best Answer

Modulo inverses
Find
\(24^{-1} \pmod{11^2}\).
That is, find the residue b  for which \(24b \equiv 1\pmod{11^2}\).
Express your answer as an integer from 0 to , inclusive.

 

\(\begin{array}{|rcll|} \hline & 24\cdot \underbrace{24^{-1}}_{=b} \equiv 1 \pmod{11^2} \\ \hline \end{array}\)

 

\(\begin{array}{rcll} \text{Let} \\ & 24\cdot 5 = 120 \equiv -1 \pmod{11^2} \\ \end{array} \)

 

\(\begin{array}{llcll} \text{square this equation: } \\ & (24\cdot 5)\cdot(24\cdot 5) &\equiv& (-1)\cdot(-1) \pmod{11^2} \\ & 24\cdot (5^2\cdot 24) &\equiv& 1 \pmod{11^2} \\ & 24\cdot 600 &\equiv& 1 \pmod{11^2} \quad | \quad 600 \equiv 116 \pmod{11^2} \\ & 24\cdot \underbrace{116}_{=(24)^{-1}=b} &\equiv& 1 \pmod{11^2} \\ \end{array} \)

 

\(\text{So $24^{-1} \pmod{11^2}=\boxed{116}$ is the multiplicative inverse to $24$ modulo $11^2$}.\)

 

laugh

heureka Aug 28, 2018

7 Online Users

avatar
avatar
avatar