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

MathCuber Jan 10, 2019

#1**+3 **

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

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

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

.Rom Jan 10, 2019

#1**+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