   Chapter 2.6, Problem 25E

Chapter
Section
Textbook Problem

Show that if n is not a prime, the cancellation law stated in Exercise 24 does not hold in ℤ n .Let p be a prime integer. Prove the following cancellation law in ℤ p : If [ a ] [ x ] = [ a ] [ y ] and [ a ] ≠ [ 0 ] , then [ x ] = [ y ] .

To determine

To prove: If n is not a prime, the cancellation law “If [a][x]=[a][y] and [a], then [x]=[y]” does not hold in n.

Explanation

Given information:

The cancellation law in p: If [a][x]=[a][y] and [a], then [x]=[y], where p be a prime integer.

Proof:

Let, n is not a prime.

Assume n=4.

So, 4={,,,}

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