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Math Help - question on squares

  1. #1
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    Given a^2 -b^2=10000000^2, for positive integers a and b. Find a, such that a+b is maximum.

    Can anyone tell me how to solve this problem.
    Last edited by mr fantastic; February 26th 2009 at 03:41 AM. Reason: Merged posts and edited
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  2. #2
    Junior Member
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    a^2 - b^2 = (a+b)(a-b) = 10000000^2 = 10^14 = (2^14)*(5^14).
    We want (a+b) to be as big as possible, which means (a-b) should be as small as possible. As we can see by the factorization of it, (a-b) must be at least 2.
    So we have a-b = 2 and a+b = (2^13)*(5^14).
    a = b + 2 --> 2b + 2 = (2^13)*(5^14) --> b + 1 = (2^12)*(5^14) --> b = (2^12)*(5^14) - 1 = 24999999999999. a = 25000000000001.
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