   Chapter 2.5, Problem 9E

Chapter
Section
Textbook Problem

Find a solution x ∈ ℤ , 0 ≤ x < n , for each of the congruences a x ≡ b   ​ ( mod   n ) in Exercises 3 − 24 .Note that in each case, a and n are relatively prime. 11 x ≡ 1   ( mod   317 )

To determine

A solution x,0x<317, for the congruence 11x1(mod317) where 11 and 317 are relatively prime.

Explanation

Given information:

11 and 317 are relatively prime.

Formula used:

1) Theorem: If a and n are relatively prime, the congruence axb(modn) has a solution x in the integers, and any two solutions in are congruent modulo n.

2) 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:

When a and n are relatively prime, the Euclidean Algorithm can be used to find a solution x to

axb(modn)

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

Solve the equations in Exercises 112 for x (mentally, if possible). ax+b=c(a0)

Finite Mathematics and Applied Calculus (MindTap Course List)

Find the derivative. Simplify where possible. 38. f(t)=1+sinht1sinht

Single Variable Calculus: Early Transcendentals, Volume I

Let h be the function defined by h(x) = x3 x2 + x + 1. Find h(5), h(0), h(a), and h(a).

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

For

Study Guide for Stewart's Multivariable Calculus, 8th 