Logo
Home Study Material Contact Search
AI Chat FREE
  1. mathematics
  2. advance math
  3. the goal of this exercise is to prove the followin

The goal of this exercise is to prove the following theorem in several steps. Theorem: Let ?...

70.2K

Verified Solution

Question

Advance Math

The goal of this exercise is to prove the following theorem inseveral steps.
Theorem: Let ? and ? be natural numbers. Then, there existunique
integers ? and ? such that ? = ?? + ? and 0 ? ? < ?.
Recall: that ? is called the quotient and ? the remainder of thedivision
of ? by ?.
(a) Let ?, ? ? Z with 0 ? ? < ?. Prove that ? divides ? if andonly if ? = 0.
(b) Use part (a) to prove the uniqueness part of the theorem. Thatis, show thatiftherearetwopairs? ,? ?Zand? ,? ?Zsatisfying?=? ?+
11221
?,0?? (c) Prove that there exist such ? and ? when ? divides ?.
(d) Prove that there exist such ? and ? when ? does not divide ? byapplying the Well-Ordering Principle to the set
? = {? ? N: ? = ? ? ?? ??? ???? ? ? Z}.

Answer & Explanation Solved by verified expert
3.6 Ratings (460 Votes)
    See Answer
Get Answers to Unlimited Questions

Join us to gain access to millions of questions and expert answers. Enjoy exclusive benefits tailored just for you!

Membership Benefits:
  • Unlimited Question Access with detailed Answers
  • Zin AI - 3 Million Words
  • 10 Dall-E 3 Images
  • 20 Plot Generations
  • Conversation with Dialogue Memory
  • No Ads, Ever!
  • Access to Our Best AI Platform: Zin AI - Your personal assistant for all your inquiries!
Become a Member

Other questions asked by students

Q

F1 plants that are medium in height are produced by crossing tall with short parents. When...

Biology
Q

1.Can i use this material for making wheel(rim) for water hose cart? 2.Which kind of procuction...

Mechanical Engineering
Q

enter your notes in MS PowerPoint 2013 O Notes O Comments 2

Biology
Q

The circuit shown consists of a switch S a battery B of emf E a...

Physics
Q

S a 6 23 A particle having charge q enters a region of uniform magnetic...

Physics
Q

Calculate the work done in the following situation A constant force F 3 5 1...

Calculus
Q

Use the formula for the nth term a of a sequence a Find the values...

Calculus
Q

A company incurred $600,000 of research and development costs in its laboratory to develop a...

Accounting
Q

There should be adequate segregation of duties between _______. A. obtaining authorization for purchase...

Accounting
Q

Isaac Inc. began operations in January 2018. For certain of its property sales, Isaac recognizes...

Accounting
Q

Smithson Mining operates a silver mine in Nevada. Acquisition, exploration, and development costs totaled $67...

Accounting
Q

Ella Inc. is considering purchasing a new milling machine. The new machine costs $298,586, plus...

Finance