Book edition 7th

Author(s) Kenneth H. Rosen

Pages 808 pages

ISBN 9780073383095

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Short Answer

Trace Algorithm 4 when it is given m = 5 , n = 11 , and b = 3 as input. That is, show all the steps Algorithm 4 uses to find 3 mod 5 .

The value is $3^{11} \mod 5 = 2$ .

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Recursive Definition

From the recursive definition,

$mpower (b, n, m) = {\begin{cases} 1 if n = 0 \\ mpower (b, n / 2, m)^{2} mod m if n even \\ [mpower (b, ⌊ n / 2 ⌋, m)^{2} mod m \cdot b mod m] mod m otherwise \end{cases}$

Step 2: Determine the recursive

Evaluate the recursive definition at n = 11 , m = 5 and b = 3.

$3^{11} \mod 5 = mpower (3, 11, 5)$

$\begin{aligned} = [mpower (3, ⌊ 11 / 2 ⌋, 5)^{2} mod 5 \cdot 3 mod 5] mod 5 \\ = [mpower (3, 5, 5)^{2} mod 5 \cdot 3] mod 5 \end{aligned}$

Determine mpower (3,5,5).

$\begin{aligned} mpower (3, 5, 5) = [mpower (3, ⌊ 5 / 2 ⌋, 5)^{2} mod 5 \cdot 3 mod 5] mod 5 \\ = [mpower (3, 2, 5)^{2} mod 5 \cdot 3] mod 5 \end{aligned}$

Determine mpower (3,2,5).

$\begin{aligned} mpower (3, 2, 5) = mpower (3, 2 / 2, 5) mod 5 \\ = mpower (3, 1, 5) mod 5 \\ = [mpower (3, ⌊ 1 / 2 ⌋, 5)^{2} mod 5 \cdot 3 mod 5] mod 5 \\ = [mpower (3, 0, 5)^{2} mod 5 \cdot 3] mod 5 \end{aligned}$

Simplify further.

$\begin{aligned} mpower (3, 2, 5) = [1^{2} mod 5 \cdot 3] mod 5 \\ = [1 \cdot 3] mod 5 \\ = 3 mod 5 \\ = 3 \end{aligned}$

Evaluate the found expression for mpower (3,5,5).

$\begin{aligned} mpower (3, 5, 5) = [mpower (3, ⌊ 5 / 2 ⌋, 5)^{2} mod 5 \cdot 3 mod 5] mod 5 \\ = [mpower (3, 2, 5)^{2} mod 5 \cdot 3] mod 5 \\ = [3^{2} mod 5 \cdot 3] mod 5 \\ = [9 mod 5 \cdot 3] mod 5 \end{aligned}$

Simplify further.

$\begin{aligned} mpower (3, 5, 5) = [4 \cdot 3] mod 5 \\ = 12 mod 5 \\ = 2 \end{aligned}$

Evaluate the found expression for mpower (3,11,5).

$\begin{aligned} 3^{11} mod 5 = mpower (3, 11, 5) \\ = [mpower (3, ⌊ 11 / 2 ⌋, 5)^{2} mod 5 \cdot 3 mod 5] mod 5 \\ = [mpower (3, 5, 5)^{2} mod 5 \cdot 3] mod 5 \\ = [2^{2} mod 5 \cdot 3] mod 5 \end{aligned}$

Simplify further.

$\begin{aligned} 3^{11} mod 5 = [4 mod 5 \cdot 3] mod 5 \\ = [4 \cdot 3] mod 5 \\ = 12 mod 5 \\ = 2 \end{aligned}$

Therefore, the value is $3^{11} \mod 5 = 2$ .

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Discrete Mathematics and its Applications

Answers without the blur.

Short Answer

Trace Algorithm 4 when it is given m = 5 , n = 11 , and b = 3 as input. That is, show all the steps Algorithm 4 uses to find 3 mod 5 .

Step by Step Solution

Step 1: Recursive Definition

Step 2: Determine the recursive

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Discrete Mathematics and its Applications

Answers without the blur.

Short Answer

Trace Algorithm 4 when it is given m = 5 , n = 11 , and b = 3 as input. That is, show all the steps Algorithm 4 uses to find 3 mod 5 .

Step by Step Solution

Step 1: Recursive Definition

Step 2: Determine the recursive

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