Results 1 to 2 of 2

Math Help - Number Theory (8)

  1. #1
    Junior Member
    Joined
    Apr 2008
    From
    Gainesville
    Posts
    68

    Number Theory (8)

    Let m and n be positive integers with (m,n) =1. Prove that each divisor d>0 of mn can be written uniquely as d1d2 where d1,d2>0, d1 divides m, d2 divides n, and (d1,d2) =1 and each such product d1d2 corresponds to a divisor d of nm.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member PaulRS's Avatar
    Joined
    Oct 2007
    Posts
    571
    Let d|(m\cdot{n}) and d=d_1\cdot{d_2}=d_3\cdot{d_4} where d_1|m,d_3|m and d_2|n,d_4|n

    we have: <br />
d_1  \cdot d_2  = \dot d_3 <br />
and since (d_2,d_3)=1 ( because (n,m)=1 ) it follows that d_1=\dot d_3 (1)

    Again <br />
d_3  \cdot d_4  = \dot d_1 <br />
and since (d_4,d_1)=1 it follows that d_3=\dot d_1 (2)

    By (1) and (2) we must have (think about the quotient) d_1=d_3 and from there d_2=d_4 which proves that the representation is unique
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Textbooks on Galois Theory and Algebraic Number Theory
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: July 8th 2011, 07:09 PM
  2. Number Theory
    Posted in the Number Theory Forum
    Replies: 3
    Last Post: May 19th 2010, 08:51 PM
  3. Number Theory
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: February 16th 2010, 06:05 PM
  4. Replies: 2
    Last Post: December 18th 2008, 06:28 PM
  5. Number theory, prime number
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: September 17th 2006, 09:11 PM

Search Tags


/mathhelpforum @mathhelpforum