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For how many integers a satisfying \(1 \le a \le 23\) is it true that \(a^{-1} \equiv a \pmod{24}\)?

 Jan 10, 2019

Best Answer 

 #1
avatar+4457 
+3

\(a^{-1}\equiv a \pmod{24} \Longleftrightarrow a^2 \equiv 1 \pmod{24}\\ a^2-1 = 24k,~k \in \mathbb{Z}\)

 

\(|\{1,5,7,11,13,17,19,23\}|=8\)

 

\(\text{We notice that these are primes such that }p^2 > 24\\ \text{There must be an algebraic reason for this and I'll see if I can dig it up}\)

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 Jan 10, 2019
 #1
avatar+4457 
+3
Best Answer

\(a^{-1}\equiv a \pmod{24} \Longleftrightarrow a^2 \equiv 1 \pmod{24}\\ a^2-1 = 24k,~k \in \mathbb{Z}\)

 

\(|\{1,5,7,11,13,17,19,23\}|=8\)

 

\(\text{We notice that these are primes such that }p^2 > 24\\ \text{There must be an algebraic reason for this and I'll see if I can dig it up}\)

Rom Jan 10, 2019

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