Results 1 to 7 of 7

Thread: application of pigeonhole principle

  1. #1
    Newbie
    Joined
    Nov 2011
    Posts
    22

    application of pigeonhole principle

    It is given 13 integers $\displaystyle c_1,c_2,...,c_{13}$ (some of them may be the same). Use pigeonhole principle to prove that there exist i and j with $\displaystyle 0<i<j<=13$ such that
    $\displaystyle c_{i+1}+c_{i+2}+...+c_j$ is divisible by 13
    for example, $\displaystyle c_4+c_5+c_6+c_7$ is divisible by 13.
    (Hint:consider the following 13 integers
    $\displaystyle n_1=c_1$
    $\displaystyle n_2=c_1+c_2$
    .
    .
    .
    $\displaystyle n_{13}=c_1+c_2+...+c_{13}$
    and their remainder when divided by 13)
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    21,782
    Thanks
    2824
    Awards
    1

    Re: application of pigeonhole principle

    Quote Originally Posted by maoro View Post
    It is given 13 integers $\displaystyle c_1,c_2,...,c_{13}$ (some of them may be the same). Use pigeonhole principle to prove that there exist i and j with $\displaystyle 0<i<j<=13$ such that
    [CENTER]$\displaystyle c_{i+1}+c_{i+2}+...+c_j$ is divisible by 13
    [LEFT]for example, $\displaystyle c_4+c_5+c_6+c_7$ is divisible by 13.
    (Hint:consider the following 13 integers
    The idea behind this proof is that if two numbers have the same remainder when divided by 13 then there difference is divisible by 13.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Nov 2011
    Posts
    22

    Re: application of pigeonhole principle

    But how this question is related to the pigeonhole principle?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Aug 2006
    Posts
    21,782
    Thanks
    2824
    Awards
    1

    Re: application of pigeonhole principle

    Quote Originally Posted by maoro View Post
    But how this question is related to the pigeonhole principle?
    Of course, we use the pigeonhole principle.
    The reminders of the of the $\displaystyle n_k$ when divided by 13 are the the pigeonholes. The $\displaystyle n_k$ are the pigeons.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    Nov 2011
    Posts
    22

    Re: application of pigeonhole principle

    I got you mean.
    Suppose $\displaystyle n_k$ is 14,then its remainder is 1 when $\displaystyle n_k$ is divided by 13
    $\displaystyle =>$ there is one pigeonhole containing more than 1 pigeon.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Newbie
    Joined
    Nov 2011
    Posts
    22

    Re: application of pigeonhole principle

    But how about if all of $\displaystyle n_i$ is not divisible by 13?
    Follow Math Help Forum on Facebook and Google+

  7. #7
    MHF Contributor

    Joined
    Aug 2006
    Posts
    21,782
    Thanks
    2824
    Awards
    1

    Re: application of pigeonhole principle

    Quote Originally Posted by maoro View Post
    But how about if all of $\displaystyle n_i$ is not divisible by 13?
    That means the zero pigeonhole some $\displaystyle n_j$ so it is divisible by 13.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Pigeonhole principle.
    Posted in the Number Theory Forum
    Replies: 0
    Last Post: Oct 9th 2010, 06:24 AM
  2. [SOLVED] Application of the Pigeonhole Principle
    Posted in the Discrete Math Forum
    Replies: 5
    Last Post: Nov 19th 2009, 05:47 PM
  3. Pigeonhole Principle
    Posted in the Discrete Math Forum
    Replies: 3
    Last Post: Oct 26th 2008, 09:09 AM
  4. Pigeonhole principle?
    Posted in the Math Topics Forum
    Replies: 2
    Last Post: Oct 16th 2008, 10:08 AM
  5. Pigeonhole principle?
    Posted in the Advanced Math Topics Forum
    Replies: 3
    Last Post: Dec 9th 2007, 08:58 AM

Search Tags


/mathhelpforum @mathhelpforum