Elementary number theory in sets

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I'm back again
So there's another problem that I can't get to prove



If we take 21 numbers randomly from $1, 2, 3, ..., 40$ then between those $21$ numbers we will be able to find two numbers, of which the smaller one will divide the bigger one



I've been reading james hein "discrete structures, logic and computability" but still can't get to think logically myself.
I would be very grateful if I could get some directions/tips/hints or anything, thanks in advance






combinatorics elementary-number-theory discrete-mathematics divisibility







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    up vote
    6
    down vote

    favorite
    1












    I'm back again
    So there's another problem that I can't get to prove



    If we take 21 numbers randomly from $1, 2, 3, ..., 40$ then between those $21$ numbers we will be able to find two numbers, of which the smaller one will divide the bigger one



    I've been reading james hein "discrete structures, logic and computability" but still can't get to think logically myself.
    I would be very grateful if I could get some directions/tips/hints or anything, thanks in advance






    combinatorics elementary-number-theory discrete-mathematics divisibility







    share|cite|improve this question

























      up vote
      6
      down vote

      favorite
      1









      up vote
      6
      down vote

      favorite
      1






      1





      I'm back again
      So there's another problem that I can't get to prove



      If we take 21 numbers randomly from $1, 2, 3, ..., 40$ then between those $21$ numbers we will be able to find two numbers, of which the smaller one will divide the bigger one



      I've been reading james hein "discrete structures, logic and computability" but still can't get to think logically myself.
      I would be very grateful if I could get some directions/tips/hints or anything, thanks in advance






      combinatorics elementary-number-theory discrete-mathematics divisibility







      share|cite|improve this question















      I'm back again
      So there's another problem that I can't get to prove



      If we take 21 numbers randomly from $1, 2, 3, ..., 40$ then between those $21$ numbers we will be able to find two numbers, of which the smaller one will divide the bigger one



      I've been reading james hein "discrete structures, logic and computability" but still can't get to think logically myself.
      I would be very grateful if I could get some directions/tips/hints or anything, thanks in advance






      combinatorics elementary-number-theory discrete-mathematics divisibility




      combinatorics elementary-number-theory discrete-mathematics divisibility






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      share|cite|improve this question













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      edited 3 hours ago









      greedoid

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      29.6k93879










      asked 3 hours ago









      Nojus Kudaba

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          1 Answer
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          Make a following sets:



          $$ A= 1,2,4,8,16,32$$
          $$ B= 3,6,12,24$$
          $$ C = 5,10,20,40$$
          $$ D = 7,14,28$$
          $$ E = 9,18,36$$
          $$F = 11,22$$
          $$G = 13,26$$
          $$H= 15,30$$
          $$I = 17,34$$
          $$J = 19,38$$
          $$K = 21,23,25,27,29,31,33,35,37,39$$



          Suppose the statement is not true. Then we take from each set A,B,...,J at most one element and if we take all elements from K we have a total of 20 elements. A contradiction.






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            1 Answer
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            active

            oldest

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            1 Answer
            1






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes








            up vote
            5
            down vote



            accepted










            Make a following sets:



            $$ A= 1,2,4,8,16,32$$
            $$ B= 3,6,12,24$$
            $$ C = 5,10,20,40$$
            $$ D = 7,14,28$$
            $$ E = 9,18,36$$
            $$F = 11,22$$
            $$G = 13,26$$
            $$H= 15,30$$
            $$I = 17,34$$
            $$J = 19,38$$
            $$K = 21,23,25,27,29,31,33,35,37,39$$



            Suppose the statement is not true. Then we take from each set A,B,...,J at most one element and if we take all elements from K we have a total of 20 elements. A contradiction.






            share|cite|improve this answer


























              up vote
              5
              down vote



              accepted










              Make a following sets:



              $$ A= 1,2,4,8,16,32$$
              $$ B= 3,6,12,24$$
              $$ C = 5,10,20,40$$
              $$ D = 7,14,28$$
              $$ E = 9,18,36$$
              $$F = 11,22$$
              $$G = 13,26$$
              $$H= 15,30$$
              $$I = 17,34$$
              $$J = 19,38$$
              $$K = 21,23,25,27,29,31,33,35,37,39$$



              Suppose the statement is not true. Then we take from each set A,B,...,J at most one element and if we take all elements from K we have a total of 20 elements. A contradiction.






              share|cite|improve this answer
























                up vote
                5
                down vote



                accepted







                up vote
                5
                down vote



                accepted






                Make a following sets:



                $$ A= 1,2,4,8,16,32$$
                $$ B= 3,6,12,24$$
                $$ C = 5,10,20,40$$
                $$ D = 7,14,28$$
                $$ E = 9,18,36$$
                $$F = 11,22$$
                $$G = 13,26$$
                $$H= 15,30$$
                $$I = 17,34$$
                $$J = 19,38$$
                $$K = 21,23,25,27,29,31,33,35,37,39$$



                Suppose the statement is not true. Then we take from each set A,B,...,J at most one element and if we take all elements from K we have a total of 20 elements. A contradiction.






                share|cite|improve this answer














                Make a following sets:



                $$ A= 1,2,4,8,16,32$$
                $$ B= 3,6,12,24$$
                $$ C = 5,10,20,40$$
                $$ D = 7,14,28$$
                $$ E = 9,18,36$$
                $$F = 11,22$$
                $$G = 13,26$$
                $$H= 15,30$$
                $$I = 17,34$$
                $$J = 19,38$$
                $$K = 21,23,25,27,29,31,33,35,37,39$$



                Suppose the statement is not true. Then we take from each set A,B,...,J at most one element and if we take all elements from K we have a total of 20 elements. A contradiction.







                share|cite|improve this answer














                share|cite|improve this answer



                share|cite|improve this answer








                edited 3 hours ago

























                answered 3 hours ago









                greedoid

                29.6k93879




                29.6k93879



























                     

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