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.
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+26367 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.
\(\begin{array}{|rcll|} \hline & 24\cdot \underbrace{24^{-1}}_{=b} \equiv 1 \pmod{11^2} \\ \hline \end{array}\)
\(\begin{array}{rcll} \text{Let} \\ & 24\cdot 5 = 120 \equiv -1 \pmod{11^2} \\ \end{array} \)
\(\begin{array}{llcll} \text{square this equation: } \\ & (24\cdot 5)\cdot(24\cdot 5) &\equiv& (-1)\cdot(-1) \pmod{11^2} \\ & 24\cdot (5^2\cdot 24) &\equiv& 1 \pmod{11^2} \\ & 24\cdot 600 &\equiv& 1 \pmod{11^2} \quad | \quad 600 \equiv 116 \pmod{11^2} \\ & 24\cdot \underbrace{116}_{=(24)^{-1}=b} &\equiv& 1 \pmod{11^2} \\ \end{array} \)
\(\text{So $24^{-1} \pmod{11^2}=\boxed{116}$ is the multiplicative inverse to $24$ modulo $11^2$}.\)
+26367 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.
\(\begin{array}{|rcll|} \hline & 24\cdot \underbrace{24^{-1}}_{=b} \equiv 1 \pmod{11^2} \\ \hline \end{array}\)
\(\begin{array}{rcll} \text{Let} \\ & 24\cdot 5 = 120 \equiv -1 \pmod{11^2} \\ \end{array} \)
\(\begin{array}{llcll} \text{square this equation: } \\ & (24\cdot 5)\cdot(24\cdot 5) &\equiv& (-1)\cdot(-1) \pmod{11^2} \\ & 24\cdot (5^2\cdot 24) &\equiv& 1 \pmod{11^2} \\ & 24\cdot 600 &\equiv& 1 \pmod{11^2} \quad | \quad 600 \equiv 116 \pmod{11^2} \\ & 24\cdot \underbrace{116}_{=(24)^{-1}=b} &\equiv& 1 \pmod{11^2} \\ \end{array} \)
\(\text{So $24^{-1} \pmod{11^2}=\boxed{116}$ is the multiplicative inverse to $24$ modulo $11^2$}.\)
