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

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

Guest Jul 29, 2018
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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.

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

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

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

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

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

heureka  Jul 30, 2018