   Chapter 2.6, Problem 20E

Chapter
Section
Textbook Problem

a. Let [ a ] ∈ ℤ n . Use mathematical induction to prove that k [ a ]   =   [ k a ] for all positive integers k b. Evaluate n [ a ] in ℤ n

(a)

To determine

To prove: k[a]=[ka] for all positive integers k.

Explanation

Given information:

[a]n and use mathematical induction.

Formula used:

1) Mathematical Induction:

The given statement Pn is true for all positive integers n if,

a. Pn is true for n=1

b. The truth of Pk always implies that Pk+1 is true.

2) Multiples in n: (k+1)[a]=k[a]+[a].

Proof:

The set of congruence classes is denoted by n.

n={,,,...,[n1]}.

Consider the statement k[a]=[ka] for all positive integers k.

By mathematical induction,

a. For k=1,

1[a]=[a] and [1a]=[a].

Thus, k[a]=[ka] for k=1.

For k=1, the statement is true

(b)

To determine

n[a] in n.

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