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Show that if a. b, k. and m are integers such that k ? 1,

Discrete Mathematics and Its Applications | 7th Edition | ISBN: 9780073383095 | Authors: Kenneth Rosen ISBN: 9780073383095 37

Solution for problem 41E Chapter 4.1

Discrete Mathematics and Its Applications | 7th Edition

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Discrete Mathematics and Its Applications | 7th Edition | ISBN: 9780073383095 | Authors: Kenneth Rosen

Discrete Mathematics and Its Applications | 7th Edition

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Problem 41E

Show that if a. b, k. and m are integers such that k ≥ 1, m ≥ 2, and a ≡ b (mod m), then ak ≡ bk(mod m).

Step-by-Step Solution:

Step 1</p>

In this problem we have to show that if a. b, k. and m are integers such that

 k ≥ 1, m ≥ 2, and a ≡ b (mod m),

 then ak ≡ bk(mod m).

Step 2</p>

ak ≡ bk(mod m) states that is congruent to bk(mod m).

Or, in other word we can say  m divides integer  and we get another integer, say quotient q.

In mathematical form we can write

When we expand  then we get,

Let, which is also an integer.

we have

Therefore we can write ak ≡ bk(mod m) as

 . ………………….(1)

Step 3 of 3

Chapter 4.1, Problem 41E is Solved
Textbook: Discrete Mathematics and Its Applications
Edition: 7
Author: Kenneth Rosen
ISBN: 9780073383095

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