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Homework Help: 68068 as difference of two squares

  1. Nov 2, 2014 #1
    1. The problem statement, all variables and given/known data
    In how many ways can 68068 be written as the difference of two squares?

    2. Relevant equations

    3. The attempt at a solution
    Let (x+a) * (x+a) -x*x =68068=2*2*7*11*13*17
    a (2x+a) =2*2*7*11*13*17
    As 2x+a is odd ⇒ a is even
    2b (2x+2b) =2*2*7*11*13*17
    b (x+b) =7*11*13*17
    x= (7*11*13*17) /b - b
    Since 7*11*13*17 has 16 factors ∴ x has 16 different values
    However, the book states 8 as the answer.
  2. jcsd
  3. Nov 2, 2014 #2


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    Homework Helper

    It looks like you need to account for the pairs (x,b) (b,x) , since those would not be distinct differences of squares.
  4. Nov 2, 2014 #3
    I guess you are right. (a, b) and (b, a) are repeated in my solution, taking pair (a,b) and (b,a) as a single pair yields 8 as the answer.
  5. Nov 2, 2014 #4


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    Gold Member

    That's not quite the reason. Remember that you had factors b and x+b. That implies a constraint.
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