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Find 2^{-1} mod{185}, as a residue modulo 185. (Give an answer between 0 and 184, inclusive.)

 Jul 29, 2018
 #1
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If a number is congruent to 2^(-1), it basically means that when multiplied by 2, it is congruent to 1 mod 185. So the answer is 93.

 Jul 29, 2018
 #2
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Find 2^{-1} mod{185}, as a residue modulo 185. (Give an answer between 0 and 184, inclusive.)

 

\(\begin{array}{rcll} \text{Let} \\ & 2\cdot 2^{-1} \equiv 1 \pmod{185} \\ \end{array} \)

 

\(\begin{array}{rcll} \text{Let} \\ & 2\cdot 92 = 184 \equiv -1 \pmod {185} \\ \end{array} \)

 

\(\begin{array}{llcll} \text{square this equation: } \\ & (2\cdot 92)(2\cdot 92) &\equiv& (-1)(-1) \pmod{185} \\ & (2) (92^2\cdot 2) &\equiv& 1 \pmod{185} \\ & (2) (16928) &\equiv& 1 \pmod{185} \quad | \quad 16928 \equiv 93 \pmod{185} \\ & (2)\underbrace{(93)}_{=(2)^{-1}} &\equiv& 1 \pmod{185} \\ \end{array}\)

 

\(\text{So $2^{-1}=\boxed{93}$ is the multiplicative inverse to $2$ modulo $185$}.\)

 

laugh

 Jul 30, 2018

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