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PS: Generic Ques

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Joined: 18 Nov 2004
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PS: Generic Ques [#permalink] New post 09 May 2005, 09:06
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I have a generic question on how to do something faster.

Say we have a eqn :

Y = (-3/5) X + 30

where 0<= X <=50 and 0<= Y <=30.......Both X,Y are integers only.

Ques is to find out how many solutions are there for this eqn i.e. how many integer (X,Y) pairs are there. I know one can enumerate and then count the number of solutions. I wud be interested to know if someone knows some trick that can make this process of counting etc go quickly.
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Re: PS: Generic Ques [#permalink] New post 09 May 2005, 09:44
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Re: PS: Generic Ques [#permalink] New post 09 May 2005, 17:11
since value of y can be only integer, there are 11 solutions because x can only be ultiples of 5 so the values of y are also 11 integers.

value of x: 0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50.
value of y: 30, 27,........................................., 0.

i know i am going to enumerate, but it's not time consuming. but about advance mathematics, i have no idea however i love it, i love it, i love it. if i get a chance to take advance mathmatics/econometrics, i will definitely take.
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 [#permalink] New post 11 May 2005, 17:04
Here is a bit of theory for you.

Diophantine equation of type 1 is

Ax + By = C

Finding one solution x0, y0 to this equation gives us all solutions.

x(t) = x0 - Bt
y(t) = y0 + At

Where t is an integer variable.

If you are given constrains m <= x < = M and n <= y <= N you know that x(t) will be decreasing with t growing and y(t) will be growing.
Compare A and B to find out who will be growing/decreasing faster and pick (x0,y0) solution accordinly so that x0 is the closed integer to M from the bottom or y0 is the closed integer to n from the top.
After that find out maximum t so that x(t) or y(t) is still with the interval of the given constrain. Count values for t, check that the other part is still in the constrain.
  [#permalink] 11 May 2005, 17:04
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