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







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  • What is the source of this puzzle?
    – Dr Xorile
    13 hours ago















up vote
1
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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
























  • What is the source of this puzzle?
    – Dr Xorile
    13 hours ago













up vote
1
down vote

favorite









up vote
1
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






share|improve this question















share|improve this question













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edited 18 hours ago









Glorfindel

12.7k34879




12.7k34879










asked 19 hours ago









Govind Prajapati

1655




1655












  • What is the source of this puzzle?
    – Dr Xorile
    13 hours ago


















  • What is the source of this puzzle?
    – Dr Xorile
    13 hours ago
















What is the source of this puzzle?
– Dr Xorile
13 hours ago




What is the source of this puzzle?
– Dr Xorile
13 hours ago










1 Answer
1






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up vote
6
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 $x_2 - x_1$.







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    1 Answer
    1






    active

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    1 Answer
    1






    active

    oldest

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    active

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    active

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    votes








    up vote
    6
    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
      6
      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
        6
        down vote



        accepted







        up vote
        6
        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 18 hours ago

























        answered 18 hours ago









        Glorfindel

        12.7k34879




        12.7k34879






























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