A community for students.

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

Loser66

  • one year ago

Show that if a, b and c are integers with c | ab, then c| (a,c)(b, c) Please, help

  • This Question is Closed
  1. Loser66
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    It is easy to prove by usual way: (a, c ) = xa + yc for some x, y in Z (b, c) = sb + tc for some s, t in Z (a,c) (b,c) = (xa + yc)(sb + tc) = xasb + xatc + ycsb + yctc = ab(xs) + c ( atx ) + c (ysb) + c ( ytc) c | ab for the first term , hence c divide the sum. Problem is : my Prof wants me to use prime factorization to prove it.

  2. Loser66
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    My attempt: \(c = \prod p_i^{m_i}\) \(ab= \prod p_i^{s_i}\) \(c|ab \implies m_i < s_i\) Then, no where to go. :(

  3. Loser66
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    @ganeshie8

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

    I've only tried the easy way...interesting how you would use it for prime factorisation

  5. Loser66
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    I don't get why c | ab, implies \(c_i \geq a_i + b_i\)

  6. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    Oops, it is a typo, fixed here : Let : \(a = \prod p_i^{a_i}\) \(b = \prod p_i^{b_i}\) \(c = \prod p_i^{c_i}\) then, \((a,c) = \prod p_i^{\min\{a_i,c_i\}}\) \((b,c) = \prod p_i^{\min\{b_i,c_i\}}\) \(c\mid ab \implies c_i \le a_i+b_i\) so, \(\min\{a_i,c_i\}+\min\{b_i,c_i\} \le a_i+b_i \ge c_i \blacksquare \)

  7. Loser66
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Got you. Thank you so much.

  8. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    np

  9. Not the answer you are looking for?
    Search for more explanations.

    • Attachments:

Ask your own question

Sign Up
Find more explanations on OpenStudy
Privacy Policy

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.