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Euclidean Linear Combination

  1. Feb 24, 2004 #1
    I need to be able to plug in appropriate x and y values for:

    154x + 260y = 4

    I guess this is done by working the euclidean algorithm backwards. But how do you do that?
     
  2. jcsd
  3. Feb 24, 2004 #2

    jamesrc

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    Gold Member

    Well, it's been like forever and a day since I did problems like this, but I think it goes something like this:

    154x + 260y = 4

    260 = (1)*154 + 106
    154 = (1)*106 + 48
    106 = (2)*48 + 10
    48 = (4)*10 + 8
    10 = (1)*8 + 2
    8 = (4)*2

    4 = 2*2 = 2*(10-8) = 2*(10-(48-4*10))
    = 10*10 - 2*48
    = 10*(106 - 2*48) - 2*48
    = 10*106 - 22*48
    = 10*(260 - 154) - 22*(154-106)
    = 10*260 - 32*154 + 22*106
    = 10*260 - 32*154 + 22*(260-154)
    = 32*260 - 54*154

    so x = -54 and y = 32
     
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