Quantcast

Got Homework?

Connect with other students for help. It's a free community.

  • across
    MIT Grad Student
    Online now
  • laura*
    Helped 1,000 students
    Online now
  • Hero
    College Math Guru
    Online now

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

Hunus Group Title

Number Theory problem

  • one year ago
  • one year ago

  • This Question is Closed
  1. Hunus Group Title
    Best Response
    You've already chosen the best response.
    Medals 0

    The division algorithm states 'For any positive integers a and b there exists a unique pair (q, r) of non-negative integers such that b = aq + r, r < a.' For the uniqueness, assume that b = aq′ + r′, where q′ and r′ are also nonnegative integers satisfying 0 ≤ r′ < a. Then aq+r = aq′+r′, implying a(q−q′) = r′−r, and so a|r′ −r. Hence |r′−r| ≥ a or |r′ −r| = 0. Because 0 ≤ r, r′ < a yields |r′ − r| < a, we are left with |r′ − r| = 0, implying r′ = r and, consequently, q′ = q.

    • one year ago
  2. Hunus Group Title
    Best Response
    You've already chosen the best response.
    Medals 0

    My question lies with the statement 'Hence |r′−r| ≥ a or |r′ −r| = 0' Why does |r′−r| ≥ a imply that |r′ −r| = 0? Is it because r must be less than a and r' - r being greater than a is invalid so it must be zero?

    • one year ago
  3. experimentX Group Title
    Best Response
    You've already chosen the best response.
    Medals 2

    a(q−q′) = r′−r consider both q's and r's to be integer, if \( q' \neq q \), you would get \( k = |q - q'| \ge 0 \)| but \( k a = |r-r'|\) means |r-r'| is some integer multiple of a .. so it follows.

    • one year ago
  4. Hunus Group Title
    Best Response
    You've already chosen the best response.
    Medals 0

    I don't quite understand how that implies that |r - r'| = 0

    • one year ago
  5. experimentX Group Title
    Best Response
    You've already chosen the best response.
    Medals 2

    since both r and r' are assumed to be positive integers which is less than 'a' .. their difference can't be greater than 'a' .. can they?

    • one year ago
  6. Hunus Group Title
    Best Response
    You've already chosen the best response.
    Medals 0

    Ahh. Thank you :) I understand now

    • one year ago
  7. experimentX Group Title
    Best Response
    You've already chosen the best response.
    Medals 2

    yw

    • one year ago
    • Attachments:

See more questions >>>

Your question is ready. Sign up for free to start getting answers.

spraguer (Moderator)
5 → View Detailed Profile

is replying to Can someone tell me what button the professor is hitting...

23

  • Teamwork 19 Teammate
  • Problem Solving 19 Hero
  • You have blocked this person.
  • ✔ You're a fan Checking fan status...

Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.

This is the testimonial you wrote.
You haven't written a testimonial for Owlfred.