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General Solutions to a Problem

  1. Oct 8, 2010 #1
    I am trying to find a general rule for finding solutions to the problem ab-a-b=x. Solving for a or b, it is apparent that 2 and 2+x always work as solutions. Some values of x, however, such as as 50, have multiple solutions. Any suggestions as to how to solve for these solutions?
     
  2. jcsd
  3. Oct 9, 2010 #2
    You can simplify the equation a bit:
    ab-a-b = x
    a(b-1)-b = x
    substitute c = b-1
    ac-c-1 = x
    c(a-1) = x+1
    substitute d=a-1
    cd = x+1
    so any two numbers whose product equals x+1 give a solution.

    for x=50:
    51 = 1*51 = 51*1 = 3*17 = 17*3 = (-1*)(-51) = (-51)*(-1) = (-3)*(-17) = (-17)*(-3)
    so there are 8 solutions: (2, 52), (52, 2), (4, 18), (18, 4), (0,-50), (-50,0), (-2, -16), (-16, -2)
     
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