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    Member Rimas's Avatar
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    pairs of positive integers which solve equation

    Find all pairs of integers x and y such that x^2-y^2=104
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    Forum Admin topsquark's Avatar
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    Quote Originally Posted by Rimas View Post
    Find all pairs of integers x and y such that x^2-y^2=104
    No doubt that ThePerfectHacker will want to weigh in on this, but a possible starting point (on a fairly long road!) would be to factor the LHS:
    (x + y)(x - y) = 104

    Now factor 104:
    1, 104
    2, 52
    4, 26
    -1, -104
    -2, -52
    -4, 26
    are all the possibilities.

    Now you have to solve simultaneous equations:
    x + y = 1
    x - y = 104

    The solution to this is x = 105/2 and y = -103/2, which aren't integers so this doesn't work. So move on to the next possibility. etc.

    I note that there are solutions for all pairs of factors where both factors are even.

    -Dan
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    Quote Originally Posted by topsquark View Post
    No doubt that ThePerfectHacker will want to weigh in on this, but a possible starting point (on a fairly long road!) would be to factor the LHS:
    (x + y)(x - y) = 104
    That is a perfectly acceptable solution.
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    Member Rimas's Avatar
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    how did you get the last part
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    Forum Admin topsquark's Avatar
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    Quote Originally Posted by topsquark View Post
    No doubt that ThePerfectHacker will want to weigh in on this, but a possible starting point (on a fairly long road!) would be to factor the LHS:
    (x + y)(x - y) = 104

    Now factor 104:
    1, 104
    2, 52
    4, 26
    -1, -104
    -2, -52
    -4, 26
    are all the possibilities.

    Now you have to solve simultaneous equations:
    x + y = 1
    x - y = 104

    The solution to this is x = 105/2 and y = -103/2, which aren't integers so this doesn't work. So move on to the next possibility. etc.

    I note that there are solutions for all pairs of factors where both factors are even.

    -Dan
    Quote Originally Posted by Rimas View Post
    how did you get the last part
    You mean the part about both factors being even? Let's solve the system of equations for a general set of factors:
    x + y = a
    x - y = b
    where a*b = 104

    Solve the top equation for y:
    y = a - x

    Then insert this value of y into the bottom equation:
    x - (a - x) = b

    2x - a = b

    x = (b + a)/2

    y = a - x = a - (a + b)/2 = (b - a)/2

    In order for x and y to be integers we need a + b and a - b to be even. Thus a and b must either both be odd or must both be even. Since a*b = 104 (an even number) at least one of a and b must be even.

    Thus both a and b must be even numbers for x and y to be integers.

    -Dan
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