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Prove that a-b=9(probablity)

  1. Apr 18, 2013 #1
    1. The problem statement, all variables and given/known data
    let E be set of natural numbers where 0<n≤100 Card E = 55
    Prove that there exists atleast 2 numbers a and b in each set where a-b=9

    2. Relevant equations

    [itex]\existsk1[/itex][itex]\in[/itex][[ ]] x1=9k1+r1 r1[itex]\in[/itex][[ ]]
    [itex]\existsk2[/itex][itex]\in[/itex][[ ]] x1=9k2+r2 r2[itex]\in[/itex][[ ]]
    .
    .
    .
    [itex]\existsk55[/itex][itex]\in[/itex][[ ]] x1=9k55+r55 r2[itex]\in[/itex][[ ]]
    (using derkil theory usually solves these kinds of problems )
    but the above didn't help me, anyway i need to make sure that i could get any numbers from 1 to 100 in every equationi tried doing this
    [itex]\existsk1[/itex][itex]\in[/itex][[1, 9 ]] x1=9k1+r1 r1[itex]\in[/itex][[1,54 ]]
    [itex]\existsk2[/itex][itex]\in[/itex][[1 , 9 ]] x1=9k2+r2 r2[itex]\in[/itex][[ 1,54 ]]
    .
    .
    .
    [itex]\existsk55[/itex][itex]\in[/itex][[1 ,9 ]] x1=9k55+r55 r2[itex]\in[/itex][[1,54 ]]
    we have 55 r and these r are all in a setB Card b = 54
    then atleast there exists ri and rj where ri = rj
    so there exists xi and xj such as xi - xj = 9(ki-kj)+ri-ri
    xi-xj=9(ki-kj)
    now this sadly dosen't get me anywhere because ki - kj must be equal to 1 or -1 .. ( i hope you guys are getting what i'm trying to do here)
    anyway i randomly noticed that 54/6 = 9 so i tried another way ..
    [itex]\existsk1[/itex][itex]\in[/itex][[ [2] , [3] ]] x1=54/k1+r1 r1[itex]\in[/itex][[ ]]
    [itex]\existsk2[/itex][itex]\in[/itex][[ [2] , [3] ]] x1=54/k2+r2 r2[itex]\in[/itex][[ ]]
    .
    .
    .
    [itex]\existsk55[/itex][itex]\in[/itex][[ [2] , [3] ]] x1=54/k55+r55 r2[itex]\in[/itex][[ ]]
    now tis way "might" work if and only if the ki and kj are different we will have
    xi-xj= 54/ki - 54/kj
    =(54ki-54kj)/kikj
    but the problem on this one is that i can't find a set which contains r's and it's card <55..
    i really need help on this problem
     
  2. jcsd
  3. Apr 18, 2013 #2

    mfb

    User Avatar
    2016 Award

    Staff: Mentor

    I don't understand your notation.

    You could consider the sets {1,10,...}, {2,11,...}, ...
    Can you calculate the maximal cardinality of a set where no two numbers have a difference of 9?
     
  4. Apr 18, 2013 #3
    Thanks I knew I was just fooling IA used the same notation in a similar exercise
     
  5. Apr 18, 2013 #4

    Mark44

    Staff: Mentor

    As posted, this is pretty much gibberish.
    Were you asking a serious question? It's to tell from the work you show.
     
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