Results 1 to 4 of 4
Like Tree3Thanks
  • 1 Post By chiro
  • 1 Post By Plato
  • 1 Post By Soroban

Math Help - How do you factor 10001 without calculator?

  1. #1
    Junior Member
    Joined
    Mar 2010
    From
    Melbourne
    Posts
    30
    Thanks
    1

    How do you factor 10001 without calculator?

    This Olympiad questions states:

    [Show that every term of the sequence 10001, 100010001, 1000100010001, ... is composite.]

    Well, I figured out this sequence is generalised as an = 10001n for all n >= 1.
    After actually doing this division in Excel(I know... I cheated), I found out the prime factorisation of 10001 = 73*137, hence proving the claim.
    If I had no access to Excel, how would you have come up with this factorisation. Any results we can make use of to get the prime factorisation?

    Thanks
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Sep 2012
    From
    Australia
    Posts
    3,838
    Thanks
    677

    Re: How do you factor 10001 without calculator?

    Hey willy0625.

    I vaguely remember results regarding 1 (mod 4) and 3 (mod 4) for prime-ness and composite numbers. Maybe you could search for those results and see if they apply in your situation.
    Thanks from topsquark
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,801
    Thanks
    1691
    Awards
    1

    Re: How do you factor 10001 without calculator?

    Quote Originally Posted by willy0625 View Post
    This Olympiad questions states:

    [Show that every term of the sequence 10001, 100010001, 1000100010001, ... is composite.]
    Note that a_0=1001 and if n\ge 1 then a_n=a_{n-1}+10^{-4(n+1)}
    Thanks from willy0625
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,809
    Thanks
    701

    Re: How do you factor 10001 without calculator?

    Hello, willy0625!

    Show that every term of the sequence 10001, 100010001, 1000100010001, ... is composite.

    Well, I figured out this sequence is generalised as an = 10001n for all n >= 1. . No!


    Not true . . .. . \begin{array}{c}10001^2 \:=\:{\bf1}000{\bf2}000{\bf1} \\ 10001^3 \:=\:{\bf1}000{\bf3}000{\bf3}000{\bf1} \\ 10001^4 \:=\:{\bf1}000{\bf4}000{\bf6}000{\bf4}000{\bf1} \end{array} . . . . Note the "binomial" pattern.


    \begin{array}{cccccc}\text{We have:} & a_1 &=& 10001 &=& 10^4 + 1 \\ & a_2 &=&100010001 &=& 10^8 + 10^4 + 1 \\ & a_3 &=&1000100010001 &=& 10^{12} + 10^8 + 10^4 + 1 \\ & \vdots && \vdots && \vdots \\ & a_n &=& 1000100010001...0001 &=& 10^{4n} + 10^{4(n-1)} +  \cdots + 10^8 + 10^4+1 \end{array}

    \text{That is: }\:a_n \;=\;\sum^n_{k=0}10^{4k}


    Want to try it again?
    Thanks from topsquark
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 2
    Last Post: May 6th 2012, 02:16 PM
  2. Replies: 0
    Last Post: July 19th 2011, 11:45 AM
  3. factor analysis , extract second factor loading
    Posted in the Advanced Statistics Forum
    Replies: 0
    Last Post: June 1st 2011, 05:17 AM
  4. factor analysis , extract second factor loading
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: May 30th 2011, 05:29 AM
  5. calculator
    Posted in the Algebra Forum
    Replies: 7
    Last Post: June 1st 2008, 09:30 PM

Search Tags


/mathhelpforum @mathhelpforum