Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 1: Q22E (page 49)

Prove or disprove: If a has an inverse modulo b, then b has an inverse modulo a.

Short Answer

Expert verified

Yes, It can be proved that ifa has an inverse modulo b, then has an inverse modulo a.

Step by step solution

01

Explain inverse modulo

Modulo is the remainder of the division , that in inverse is added with quotient is called inverse modulo.

02

Prove the given statement

Consider that if ahas an inverse modulo bThen gcda,b≡1and ∃x,y∈c,such that ax+by=1.

Thus x is one of the inverse elements of amodulo b.

By symmetry, y is one of the inverse elements of b modulo a.

Therefore, it is proved that if a has an inverse modulo b, then b has an inverse modulo a.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Give an efficient algorithm to compute the least common multiple of two n-bit numbers x and y, that is, the smallest number divisible by bothx and y. What is the running time of your algorithm as a function of n?

Suppose you want to compute the nth Fibonacci number Fn , modulo an integer p. Can you find an efficient way to do this?

A positive integer N is a power if it is of the formqk , where q,role="math" localid="1658399000008" k are positive integers and k>1.

(a) Give an efficient algorithm that takes as input a number and determines whether it is a square, that is, whether it can be written asq2 for some positive integer q. What is the running time of your algorithm?

(b) Show that if N=qk (with role="math" localid="1658399171717" N,q , andk all positive integers), then either role="math" localid="1658399158890" k≤logNorN=1.

(c) Give an efficient algorithm for determining whether a positive integerN is a power. Analyze its running time.

Show that log(n!)=θ(nlogn)

(Hint: To show an upper bound, compare (n!) with nn. To show a lower bound, compare it with (n2)n2).

Consider the problem of computing N!=1·2·3···N.

(a) If Nis an role="math" localid="1658397956489" n-bit number, how many bits long is N!, approximately ( inΘ(·)form)?

(b) Give an algorithm to compute N!and analyze its running time.
See all solutions

Recommended explanations on Computer Science Textbooks

Computer Programming

Read Explanation

Data Representation in Computer Science

Read Explanation

Computer Organisation and Architecture

Read Explanation

Data Structures

Read Explanation

Algorithms in Computer Science

Read Explanation

Databases

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free