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.)
Using the Euclidean Algorith, $11^{-1} \equiv 269 \pmod{508}$.
Can you further explain your thinking?
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, $11^{-1} \equiv 3\cdot 77 \equiv 231 \pmod{508}$.
Given that 33−1≡77(mod508),find 11−1(mod508) as a residue modulo 508.(Give an answer between 0 and 507, inclusive.)
33−1=133≡77(mod508)133=13∗11≡77(mod508)13∗11≡77(mod508)|∗333∗11≡3∗77(mod508)111≡231(mod508)111=11−1≡231(mod508)
