Results 1 to 4 of 4

Thread: relatively prime integers....

  1. #1
    Newbie
    Joined
    Mar 2008
    Posts
    7

    relatively prime integers....

    Hello everbody...

    Let b and c be relatively prime integers, and suppose a is an integer that is
    divisible by both b and c. Prove that bc|a.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member Aryth's Avatar
    Joined
    Feb 2007
    From
    USA
    Posts
    666
    Thanks
    2
    Awards
    1
    If a is divisible by both b and c, we have:

    $\displaystyle bk = a$

    $\displaystyle cj = a$

    $\displaystyle b = a/k$

    $\displaystyle c = a/j$

    Therefore:

    $\displaystyle bc = a/jk$

    $\displaystyle jk = i$

    $\displaystyle bc(i) = a$

    Thus:

    $\displaystyle bc | a$
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Mar 2008
    Posts
    7
    Quote Originally Posted by Aryth View Post
    If a is divisible by both b and c, we have:

    $\displaystyle bk = a$

    $\displaystyle cj = a$

    $\displaystyle b = a/k$

    $\displaystyle c = a/j$

    Therefore:

    ***$\displaystyle bc = a/jk$***

    $\displaystyle jk = i$

    $\displaystyle bc(i) = a$

    Thus:

    $\displaystyle bc | a$
    ummm....$\displaystyle bc = a^2/jk$. how does that work?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Senior Member JaneBennet's Avatar
    Joined
    Dec 2007
    Posts
    293
    b and c divide a $\displaystyle \Rightarrow$ $\displaystyle a=hb=kc$ for some integers h and k.

    b and c are relatively prime means there are integers $\displaystyle p,q$ such that $\displaystyle pb+qc=1$.

    Now multiply through by a.

    $\displaystyle pba+qca=a$ $\displaystyle \Rightarrow$ $\displaystyle a=pb(kc)+qc(hb)=(pk+qh)bc$.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Relatively Prime Set of Integers
    Posted in the Number Theory Forum
    Replies: 3
    Last Post: May 15th 2010, 12:32 AM
  2. Relatively prime integers
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: Apr 26th 2010, 08:49 AM
  3. Replies: 3
    Last Post: Oct 21st 2009, 07:58 PM
  4. Relatively prime quadratic integers
    Posted in the Number Theory Forum
    Replies: 3
    Last Post: Oct 27th 2008, 11:11 PM
  5. pairwise relatively prime integers
    Posted in the Number Theory Forum
    Replies: 4
    Last Post: Jul 17th 2006, 10:10 AM

Search Tags


/mathhelpforum @mathhelpforum