Modular Inverses Help

Question

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

MathCuber Jan 10, 2019

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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

Rom · Answer 1 · 2019-01-10T01:14:56

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

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

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

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