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# Modulo inverses

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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

#1
+22188
+4

Modulo inverses
Find
$$24^{-1} \pmod{11^2}$$.
That is, find the residue b  for which $$24b \equiv 1\pmod{11^2}$$.

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

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

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

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

Aug 28, 2018

#1
+22188
+4

Modulo inverses
Find
$$24^{-1} \pmod{11^2}$$.
That is, find the residue b  for which $$24b \equiv 1\pmod{11^2}$$.

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

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

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

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

heureka Aug 28, 2018