   Chapter 2.5, Problem 43E

Chapter
Section
Textbook Problem

In the congruences a x ≡ b ( m o d   n ) in Exercises 40 − 53 , a and n may not be relatively prime. Use the results in Exercises 38 and 39 to determine whether there are solutions. If there are, find d incongruent solutions modulo n . 8 x ≡ 66   ( m o d  78 )

To determine

Whether there are solutions of congruence 8x66(mod78). If there are, find d incongruent solutions modulo 78.

Explanation

Formula used:

i) For d=(a,n), where n>1 if there is a solution to axb(modn), then d divides b.

ii) Suppose that n>1 and that d=(a,n) is divisor of b. Let a=a0d,b=b0d and n=n0d, where a0,b0,n0 are integers the following statements are hold:

a) axb(modn), if and only if a0xb0(modn0).

b) If x1 and x2 are any two solutions of a0xb0(modn0), then it follows that x1x2(modn0).

c) Let x1 be a fixed solution to a0xb0(modn0), and each of the d integers in the list

x1,x1+n0,x1+2n0,x1+(d1)n0 is a solution to axb(modn).

d) No two of the solutions listed in part c) are congruent modulo n.

e) Any solution to axb(modn) congruent to one of the numbers listed in part c).

iii) The Euclidean Algorithm:

a=bq0+r1,0r1<bb=r1q1+r2,0r2<r1r1=r2q2+r3,0r3<r2rk=rk+1qk+1+rk+2,0rk+2<rk+1

Since the integers r1,r2,,rk+2 are decreasing and are all non-negative, there is a smallest integer n, such that rn+1=0: rn1=rnqn+rn+1,0=rn+1.

Explanation:

Given 8x66(mod78)

Comparing 8x66(mod78) with axb(modn).

Then, a=8,n=78,b=66.

Now, d=(a,n)=(8,78)=2.

By using result, for d=(a,n) where n>1, if there is a solution to axb(modn), then d divides b

Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

Find an equation of the horizontal line that passes through (4, 3).

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

Differentiate the functions in Problems 3-20. 17.

Mathematical Applications for the Management, Life, and Social Sciences

JOB ASSIGNMENT S s brokers if each broker handles 20 inquiries?

Finite Mathematics for the Managerial, Life, and Social Sciences

True or False: converges mean exists.

Study Guide for Stewart's Multivariable Calculus, 8th

Evaluate .

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th 