Explanation Given information: The given expression is 89 307 ( mod 713 ) . Formula used: Division algorithm: Let a be an integer and d be a positive integer. Then, there are unique integers q and r with 0 ≤ r < d such that a = d q + r , here, q is the quotient and r is the remainder. So, q = a div d and r = a mod d . Know that the technique is x m ( mod y ) = [ x ( mod y ) ] m ( mod y ) and a b mod n = [ ( a mod n ) ( b mod n ) ] ( mod n ) . Calculation: Write the 307 as a sum of powers of 2. 307 = 256 + 32 + 16 + 2 + 1 First, find the values of 89 1 ( mod 713 ) , 89 2 ( mod 713 ) , 89 4 ( mod 713 ) , 89 8 ( mod 713 ) , 89 16 ( mod 713 ) , 89 32 ( mod 713 ) , 89 64 ( mod 713 ) , 89 128 ( mod 713 ) and 89 256 ( mod 713 ) . To find the value of 89 1 ( mod 713 ) , use the formula x 2 ( mod y ) = [ x ( mod y ) ] 2 ( mod y ) . 89 1 ( mod 713 ) = 89 ( mod 713 ) = 89 Thus, the value of 89 1 ( mod 713 ) is 89. To find the value of 89 2 ( mod 713 ) , use the formula x 2 ( mod y ) = [ x ( mod y ) ] 2 ( mod y ) . 89 2 ( mod 713 ) = 7921 ( mod 713 ) [ 7921 = 11 ⋅ 713 + 78 ] = 78 Thus, the value of 89 2 ( mod 713 ) is 78. To find the value of 89 4 ( mod 713 ) , use the formula x 2 ( mod y ) = [ x ( mod y ) ] 2 ( mod y ) . 89 4 ( mod 713 ) = ( 89 2 ) 2 ( mod 713 ) = [ 89 2 ( mod 713 ) ] 2 ( mod 713 ) = 78 2 ( mod 713 ) Simplify the above equation. 78 2 ( mod 713 ) = 6084 ( mod 713 ) = 380 Thus, the value of 89 4 ( mod 713 ) is 380. To find the value of 89 8 ( mod 713 ) , use the formula x 2 ( mod y ) = [ x ( mod y ) ] 2 ( mod y ) . 89 8 ( mod 713 ) = ( 89 4 ) 2 ( mod 713 ) = [ 89 4 ( mod 713 ) ] 2 ( mod 713 ) = ( 380 ) 2 ( mod 713 ) Simplify the above equation. ( 380 ) 2 ( mod 713 ) = 144400 ( mod 713 ) [ 144400 = 202 ⋅ 713 + 374 ] = 374 Thus, the value of 89 8 ( mod 713 ) is 374. To find the value of 89 16 ( mod 713 ) , use the formula x 2 ( mod y ) = [ x ( mod y ) ] 2 ( mod y ) . 89 16 ( mod 713 ) = ( 89 8 ) 2 ( mod 713 ) = [ 89 8 ( mod 713 ) ] 2 ( mod 713 ) = ( 374 ) 2 ( mod 713 ) Simplify the above equation. ( 374 ) 2 ( mod 713 ) = 139876 ( mod 713 ) [ 139876 = 196 ⋅ 713 + 128 ] = 128 Thus, the value of 89 16 ( mod 713 ) is 128. To find the value of 89 32 ( mod 713 ) , use the formula x 2 ( mod y ) = [ x ( mod y ) ] 2 ( mod y )

5th Edition

ISBN: 9781337694193

Author: EPP, Susanna S.

Publisher: Cengage Learning,

See similar textbooks

Concept explainers

Set Theory

Every field in the present world involves a set. The theory of sets was developed by German mathematician Georg Cantor while working on the “problems of trigonometric series”.

Question

Chapter 8.4, Problem 17ES

To determine

To calculate:

The value of 89307(mod713).

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 8.4, Problem 16ES

Chapter 8.4, Problem 18ES

Students have asked these similar questions

Give an example of a graph with at least 3 vertices that has exactly 2 automorphisms(one of which is necessarily the identity automorphism). Prove that your example iscorrect.

3. [10 marks] Let Go (Vo, Eo) and G₁ = (V1, E1) be two graphs that ⚫ have at least 2 vertices each, ⚫are disjoint (i.e., Von V₁ = 0), ⚫ and are both Eulerian. Consider connecting Go and G₁ by adding a set of new edges F, where each new edge has one end in Vo and the other end in V₁. (a) Is it possible to add a set of edges F of the form (x, y) with x € Vo and y = V₁ so that the resulting graph (VUV₁, Eo UE₁ UF) is Eulerian? (b) If so, what is the size of the smallest possible F? Prove that your answers are correct.

Let T be a tree. Prove that if T has a vertex of degree k, then T has at least k leaves.

Chapter 8 Solutions

Discrete Mathematics With Applications

Ch. 8.1 - If R is a relation from A to B, xA , and yB , the...Ch. 8.1 - Prob. 2TY Ch. 8.1 - Prob. 3TY Ch. 8.1 - Prob. 4TY Ch. 8.1 - If R is a relation on a set A, the directed graph...Ch. 8.1 - As in Example 8.1.2, the congruence modulo 2...Ch. 8.1 - Prove that for all integers m and n,m-n is even...Ch. 8.1 - The congruence modulo 3 relation, T, is defined...Ch. 8.1 - Define a relation P on Z as follows: For every...Ch. 8.1 - Prob. 5ES

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, subject and related others by exploring similar questions and additional content below.

The value of 89 307 ( mod 713 ) .

Solution Summary: The author explains how to calculate the value of 89307(mathrmmod713).

Concept explainers

To calculate:

The value of 89307(mod713).

Want to see the full answer?

Chapter 8 Solutions

The value of 89 307 ( mod 713 ) .

Solution Summary: The author explains how to calculate the value of 89307(mathrmmod713).

Concept explainers

To calculate: The value of 89307(mod713).

Want to see the full answer?

Chapter 8 Solutions

To calculate:

The value of 89307(mod713).