Results 1 to 3 of 3

Math Help - Divisibility proof

  1. #1
    rf0
    rf0 is offline
    Newbie
    Joined
    Sep 2008
    Posts
    2

    Divisibility proof

    gcd(a,b) = 1 and a|m and b|m. How to prove ab|m without using unique-prime-factorization theorem?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Moo
    Moo is offline
    A Cute Angle Moo's Avatar
    Joined
    Mar 2008
    From
    P(I'm here)=1/3, P(I'm there)=t+1/3
    Posts
    5,618
    Thanks
    6
    Hello,
    Quote Originally Posted by rf0 View Post
    gcd(a,b) = 1 and a|m and b|m. How to prove ab|m without using unique-prime-factorization theorem?
    a \mid m \implies m=am' ~,~ \text{where } m' \in \mathbb{Z}

    b \mid m \implies m=bm'' ~,~ \text{where } m'' \in \mathbb{Z}

    Therefore am'=bm'' and hence a divides bm''

    By Gauss theorem, if \text{gcd}(a,b)=1 and a \mid bc, then a \mid c
    So here, we can conclude that a \mid m''.
    So we can write m''=am''' ~,~ \text{where } m''' \in \mathbb{Z}

    So m=bm''=b(am''')=(ab)m'''

    Therefore ab \mid m \quad \quad \square
    Follow Math Help Forum on Facebook and Google+

  3. #3
    rf0
    rf0 is offline
    Newbie
    Joined
    Sep 2008
    Posts
    2
    Thanks a bunch, I understand it now
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Divisibility proof
    Posted in the Algebra Forum
    Replies: 7
    Last Post: June 10th 2010, 10:47 AM
  2. divisibility proof
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: January 24th 2010, 08:45 AM
  3. Divisibility Proof
    Posted in the Pre-Calculus Forum
    Replies: 4
    Last Post: March 16th 2009, 06:11 PM
  4. Proof of divisibility
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: October 13th 2008, 01:53 PM
  5. divisibility proof
    Posted in the Number Theory Forum
    Replies: 2
    Last Post: December 2nd 2007, 10:39 AM

Search Tags


/mathhelpforum @mathhelpforum