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Homework Help: A little help

  1. Feb 19, 2008 #1
    1. The problem statement, all variables and given/known data

    I don't know where to start on this question - could someone please point me in the right direction so i can look up the method.

    Q: Find integers x, y such that 45x + 63y = 99. Can we find integers s, t such that 45s +65t = 80? Either find them or prove they cannot exist.
  2. jcsd
  3. Feb 19, 2008 #2


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    What have you tried?
  4. Feb 19, 2008 #3
    when x = 5 and y = -2 , but that was just randomly trying things. is there any method or rule to this?
  5. Feb 19, 2008 #4
    Look up the Euclidean algorithm. Also, if there are common factors in all the terms, it might be in your best interest to factor them out.
  6. Feb 19, 2008 #5
    these equations are just a equation of a linear line! both of them; so you can just plot the line and say you will have infinity answers for x and y which have to be on the line. considering both lines: of course these can encounter with each other and as they are not parallel their is exactly one point that is in their intercept (if you deal with x, y).
  7. Feb 19, 2008 #6
    The equations [itex]a\,x+b\,y=c[/itex] are called Linear Diophantine Equations. If you acn find a particular solution [itex](x,y)=(x_o,y_o)[/itex] then you can find the general solution by writting [itex]x=x_o+\lambda\,t,\,y=y_o+\mu\,t[/itex]. Plugging these to the original equation you determinate the values of [itex]\lambda,\,\mu[/itex].

    You can easily proof that if [itex]c[/itex] is not divisible by [itex]gcd(a,b)[/itex] then there is no solution.
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