Book edition 7th

Author(s) Kenneth H. Rosen

Pages 808 pages

ISBN 9780073383095

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Short Answer

Show that every positive integer can be represented uniquely as the sum of distinct powers of 2 . [Hint: Consider binary expansions of integers.]

Every positive integer can be represented uniquely as the sum of distinct powers

Of 2.

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1

Let b be an integer greater than 1.

By theorem 1, every integer n can then be expressed uniquely in the form :

$n = a_{n} b^{k} + a_{k - 1} b^{k - 1} + \dots + a_{1} b + a_{0}$

In this case , we have base 2. Thus there then exist unique values

such that :

$n = a_{k} \cdot 2^{k} + a_{k - 1} \cdot 2^{k - 1} + \dots + a_{1} \cdot 2 + a_{0}$

This then means that every positive integer n can be represented uniquely as the

sum of distinct powers of 2. Moreover, the coefficient $a_{k}, a_{k - 1}, \dots, a_{0}$ are given by the

binary representation of n .

Step 2

When $a_{i} = 1$ then x is first multiplied by the power and reduced modulo 645

Then on each iteration the power is multiplied by itself and reduced modulo 645.

$i = 0 since a_{0} = 0$

x = 1

$power = 7^{2} mod 645 = 49 mod 645 = 49$

$i = 1 {Sincea}_{1} = 0 :$

x = 1

$p o w e r = 49^{2} mod 645 = 2401 mod 645 = 466$

i = 2 {Sincea}_{2} = 1

$x = 1.466 = 466$

$power = 466^{2} mod 645 = 217156 mod 645 = 436$

$i = 3 {Sincea}_{3} = 0$

$x = 466$

role="math" localid="1668514191719"

power = 436^{2} mod 645 = 190096 mod 645 = 466

$i = 5 {Sincea}_{5} = 0$

$x = 466 power = 436^{2} mod 645 = 190096 mod 645 = 466 i = 6 {Sincea}_{6} = 0 x = 466 power = 466^{2} mod 645 = 217156 mod 645 = 436$

Step 3

$i = 7 {Since}_{7} = 1 : x = 466 \cdot 436 mod 645 = 203176 mod 645 = 1 power = 436^{2} mo d 645 = 190096 mod 645 = 466 i = 8 {Sincea}_{8} = 0 : x = 1 power = 466^{2} mod 645 = 217156 mod 645 = 436 i = 9 {sincea}_{9} = 1 : x = 1.436 mod 645 = 436 mod 645 = 436 power = 436^{2} mod 645 = 190096 mod 645 = 466 return x = 436$

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Discrete Mathematics and its Applications

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Show that every positive integer can be represented uniquely as the sum of distinct powers of 2 . [Hint: Consider binary expansions of integers.]

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