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# NEED HELP QUICKLY

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Given that $33^{-1} \equiv 77$ (mod $508$), find $11^{-1}$ (mod $508$) as a residue modulo 508. (Give an answer between 0 and 507, inclusive.)

Jun 16, 2021

#1
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Using the Euclidean Algorith, $11^{-1} \equiv 269 \pmod{508}$.

Jun 16, 2021
#2
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Can you further explain your thinking?

Guest Jun 16, 2021
#3
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By definition $33^{-1} \equiv 77 \pmod{508}$ means $33\cdot 77 \equiv 1 \pmod{508}$.  Thus $11\cdot 3\cdot 77 \equiv 1 \pmod{508}$.  Hence, $11^{-1} \equiv 3\cdot 77 \equiv 231 \pmod{508}$.

Jun 16, 2021
#4
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Given that $$33^{-1} \equiv 77 \pmod{508}$$,
find $$11^{-1}\pmod{508}$$ as a residue modulo $$508$$.
(Give an answer between 0 and 507, inclusive.)

$$\begin{array}{|rcll|} \hline 33^{-1}= \dfrac{1}{33} & \equiv& 77 \pmod{508} \\\\ \dfrac{1}{33} = \dfrac{1}{3*11} & \equiv& 77 \pmod{508} \\\\ \dfrac{1}{3*11} & \equiv& 77 \pmod{508} \quad | \quad * 3 \\\\ \dfrac{3}{3*11} & \equiv& 3*77 \pmod{508} \\\\ \dfrac{1}{11} & \equiv & 231 \pmod{508} \\\\ \dfrac{1}{11}=\mathbf{ 11^{-1} } & \equiv & \mathbf{ 231 \pmod{508} } \\ \hline \end{array}$$

Jun 16, 2021