Will two kangaroos ever meet after making same number of jumps?











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There are two kangaroos on an x-axis ready to jump in the positive direction (i.e, toward positive infinity). The first kangaroo starts at location $x_1$ and moves at a rate of $v_1$ meters per jump. The second kangaroo starts at location $x_2$ and moves at a rate of $v_2$ meters per jump. Given the starting locations and movement rates for each kangaroo, can you determine if they'll ever land at the same location at the same time?



Input Format



A single line of four space-separated integers denoting the respective values of $x_1$, $v_1$, $x_2$, and $v_2$.



Constraints




  1. $0 le x_1 < x_2$

  2. $1 le v_1$

  3. $1 le v_2$


Output Format



Print YES if they can land on the same location at the same time; otherwise, print NO.



Note: The two kangaroos must land at the same location after making the same number of jumps.




Sample Input 0



0 3 4 2



Sample Output 0



YES



Explanation 0



The two kangaroos jump through the following sequence of locations:




  1. 0 3 6 9 12

  2. 4 6 8 10 12


Thus, the kangaroos meet after 4 jumps and we print YES.




Sample Input 1



0 2 5 3



Sample Output 1



NO



Explanation 1



The second kangaroo has a starting location that is ahead (further to the right) of the first kangaroo's starting location (i.e., $x_2 > x_1$). Because the second kangaroo moves at a faster rate (meaning $v_2 > v_1$) and is already ahead of the first kangaroo, the first kangaroo will never be able to catch up. Thus, we print NO.




Note: I searched for the answer and got this puzzle here but without answer :( so had to ask.






mathematics







share|improve this question




























    up vote
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    down vote

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    There are two kangaroos on an x-axis ready to jump in the positive direction (i.e, toward positive infinity). The first kangaroo starts at location $x_1$ and moves at a rate of $v_1$ meters per jump. The second kangaroo starts at location $x_2$ and moves at a rate of $v_2$ meters per jump. Given the starting locations and movement rates for each kangaroo, can you determine if they'll ever land at the same location at the same time?



    Input Format



    A single line of four space-separated integers denoting the respective values of $x_1$, $v_1$, $x_2$, and $v_2$.



    Constraints




    1. $0 le x_1 < x_2$

    2. $1 le v_1$

    3. $1 le v_2$


    Output Format



    Print YES if they can land on the same location at the same time; otherwise, print NO.



    Note: The two kangaroos must land at the same location after making the same number of jumps.




    Sample Input 0



    0 3 4 2



    Sample Output 0



    YES



    Explanation 0



    The two kangaroos jump through the following sequence of locations:




    1. 0 3 6 9 12

    2. 4 6 8 10 12


    Thus, the kangaroos meet after 4 jumps and we print YES.




    Sample Input 1



    0 2 5 3



    Sample Output 1



    NO



    Explanation 1



    The second kangaroo has a starting location that is ahead (further to the right) of the first kangaroo's starting location (i.e., $x_2 > x_1$). Because the second kangaroo moves at a faster rate (meaning $v_2 > v_1$) and is already ahead of the first kangaroo, the first kangaroo will never be able to catch up. Thus, we print NO.




    Note: I searched for the answer and got this puzzle here but without answer :( so had to ask.






    mathematics







    share|improve this question


























      up vote
      2
      down vote

      favorite









      up vote
      2
      down vote

      favorite











      There are two kangaroos on an x-axis ready to jump in the positive direction (i.e, toward positive infinity). The first kangaroo starts at location $x_1$ and moves at a rate of $v_1$ meters per jump. The second kangaroo starts at location $x_2$ and moves at a rate of $v_2$ meters per jump. Given the starting locations and movement rates for each kangaroo, can you determine if they'll ever land at the same location at the same time?



      Input Format



      A single line of four space-separated integers denoting the respective values of $x_1$, $v_1$, $x_2$, and $v_2$.



      Constraints




      1. $0 le x_1 < x_2$

      2. $1 le v_1$

      3. $1 le v_2$


      Output Format



      Print YES if they can land on the same location at the same time; otherwise, print NO.



      Note: The two kangaroos must land at the same location after making the same number of jumps.




      Sample Input 0



      0 3 4 2



      Sample Output 0



      YES



      Explanation 0



      The two kangaroos jump through the following sequence of locations:




      1. 0 3 6 9 12

      2. 4 6 8 10 12


      Thus, the kangaroos meet after 4 jumps and we print YES.




      Sample Input 1



      0 2 5 3



      Sample Output 1



      NO



      Explanation 1



      The second kangaroo has a starting location that is ahead (further to the right) of the first kangaroo's starting location (i.e., $x_2 > x_1$). Because the second kangaroo moves at a faster rate (meaning $v_2 > v_1$) and is already ahead of the first kangaroo, the first kangaroo will never be able to catch up. Thus, we print NO.




      Note: I searched for the answer and got this puzzle here but without answer :( so had to ask.






      mathematics







      share|improve this question















      There are two kangaroos on an x-axis ready to jump in the positive direction (i.e, toward positive infinity). The first kangaroo starts at location $x_1$ and moves at a rate of $v_1$ meters per jump. The second kangaroo starts at location $x_2$ and moves at a rate of $v_2$ meters per jump. Given the starting locations and movement rates for each kangaroo, can you determine if they'll ever land at the same location at the same time?



      Input Format



      A single line of four space-separated integers denoting the respective values of $x_1$, $v_1$, $x_2$, and $v_2$.



      Constraints




      1. $0 le x_1 < x_2$

      2. $1 le v_1$

      3. $1 le v_2$


      Output Format



      Print YES if they can land on the same location at the same time; otherwise, print NO.



      Note: The two kangaroos must land at the same location after making the same number of jumps.




      Sample Input 0



      0 3 4 2



      Sample Output 0



      YES



      Explanation 0



      The two kangaroos jump through the following sequence of locations:




      1. 0 3 6 9 12

      2. 4 6 8 10 12


      Thus, the kangaroos meet after 4 jumps and we print YES.




      Sample Input 1



      0 2 5 3



      Sample Output 1



      NO



      Explanation 1



      The second kangaroo has a starting location that is ahead (further to the right) of the first kangaroo's starting location (i.e., $x_2 > x_1$). Because the second kangaroo moves at a faster rate (meaning $v_2 > v_1$) and is already ahead of the first kangaroo, the first kangaroo will never be able to catch up. Thus, we print NO.




      Note: I searched for the answer and got this puzzle here but without answer :( so had to ask.






      mathematics




      mathematics






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      edited 5 hours ago









      Glorfindel

      12.7k34879




      12.7k34879










      asked 5 hours ago









      Govind Prajapati

      1675




      1675






















          1 Answer
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          accepted










          They'll meet if and only if




          $v_1 > v_2$ (so that kangaroo 1 catches up)




          and




          $v_1 - v_2 | x_2 - x_1$, here | means 'is a divisor of'.




          Why?




          After $n in Bbb{N}$ jumps, kangaroo 1 will be at position $x_1 + n v_1$ and kangaroo 2 at $x_2 + n v_2$. Now, if
          $$x_1 + n v_1 = x_2 + n v_2$$
          $$n v_1 - n v_2 = x_2 - x_1$$
          $$n (v_1 - v_2) = x_2 - x_1$$
          $$n = frac{x_2 - x_1}{v_1 - v_2}$$
          This fraction is an integer if and only if $v_1 - v_2$ divides $x_2 - x_1$.







          share|improve this answer























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            1 Answer
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            active

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            1 Answer
            1






            active

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            active

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            up vote
            4
            down vote



            accepted










            They'll meet if and only if




            $v_1 > v_2$ (so that kangaroo 1 catches up)




            and




            $v_1 - v_2 | x_2 - x_1$, here | means 'is a divisor of'.




            Why?




            After $n in Bbb{N}$ jumps, kangaroo 1 will be at position $x_1 + n v_1$ and kangaroo 2 at $x_2 + n v_2$. Now, if
            $$x_1 + n v_1 = x_2 + n v_2$$
            $$n v_1 - n v_2 = x_2 - x_1$$
            $$n (v_1 - v_2) = x_2 - x_1$$
            $$n = frac{x_2 - x_1}{v_1 - v_2}$$
            This fraction is an integer if and only if $v_1 - v_2$ divides $x_2 - x_1$.







            share|improve this answer



























              up vote
              4
              down vote



              accepted










              They'll meet if and only if




              $v_1 > v_2$ (so that kangaroo 1 catches up)




              and




              $v_1 - v_2 | x_2 - x_1$, here | means 'is a divisor of'.




              Why?




              After $n in Bbb{N}$ jumps, kangaroo 1 will be at position $x_1 + n v_1$ and kangaroo 2 at $x_2 + n v_2$. Now, if
              $$x_1 + n v_1 = x_2 + n v_2$$
              $$n v_1 - n v_2 = x_2 - x_1$$
              $$n (v_1 - v_2) = x_2 - x_1$$
              $$n = frac{x_2 - x_1}{v_1 - v_2}$$
              This fraction is an integer if and only if $v_1 - v_2$ divides $x_2 - x_1$.







              share|improve this answer

























                up vote
                4
                down vote



                accepted







                up vote
                4
                down vote



                accepted






                They'll meet if and only if




                $v_1 > v_2$ (so that kangaroo 1 catches up)




                and




                $v_1 - v_2 | x_2 - x_1$, here | means 'is a divisor of'.




                Why?




                After $n in Bbb{N}$ jumps, kangaroo 1 will be at position $x_1 + n v_1$ and kangaroo 2 at $x_2 + n v_2$. Now, if
                $$x_1 + n v_1 = x_2 + n v_2$$
                $$n v_1 - n v_2 = x_2 - x_1$$
                $$n (v_1 - v_2) = x_2 - x_1$$
                $$n = frac{x_2 - x_1}{v_1 - v_2}$$
                This fraction is an integer if and only if $v_1 - v_2$ divides $x_2 - x_1$.







                share|improve this answer














                They'll meet if and only if




                $v_1 > v_2$ (so that kangaroo 1 catches up)




                and




                $v_1 - v_2 | x_2 - x_1$, here | means 'is a divisor of'.




                Why?




                After $n in Bbb{N}$ jumps, kangaroo 1 will be at position $x_1 + n v_1$ and kangaroo 2 at $x_2 + n v_2$. Now, if
                $$x_1 + n v_1 = x_2 + n v_2$$
                $$n v_1 - n v_2 = x_2 - x_1$$
                $$n (v_1 - v_2) = x_2 - x_1$$
                $$n = frac{x_2 - x_1}{v_1 - v_2}$$
                This fraction is an integer if and only if $v_1 - v_2$ divides $x_2 - x_1$.








                share|improve this answer














                share|improve this answer



                share|improve this answer








                edited 5 hours ago

























                answered 5 hours ago









                Glorfindel

                12.7k34879




                12.7k34879






























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