   Chapter 2.6, Problem 22E

Chapter
Section
Textbook Problem

Let p be a prime integer. Prove that [ 1 ]  and  [ p − 1 ] are the only elements in ℤ p that are their own multiplicative inverses.

To determine

To prove:  and [p1] are the only elements in p that are their own multiplicative inverses.

Explanation

Given information:

p is a prime integer.

Formula used:

Theorem: Substitution Properties:

Suppose ab(modn) and cd(modn). Then a+cb+d(modn) and acbd(modn).

Proof:

p denotes the set of congruence classes.

p={,,,...,[p1]}.

Let, [a]p be any element such that [a] has their own multiplicative inverses

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