Tag Archives: Squares

Each of a+b,a+c,a+d,b+c,b+d,c+d,a+b+c+d is a square

    Find four distinct integers   a, b, c, d   such that a + b a + c a + d b + c b + d c + d a + b + c + d are … Continue reading

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Rational numbers (x,y); each of x – 1 and x + 1 is a square

    Solve,     Similarly for,     Solution: Put we get,     and                                      

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Each of a-b, a+n, b+n, a+b+n is a square

    To find two numbers,     and   ,   whose difference,     is a square and such that each,     and   ,   is a square and their sum,   ,   is a … Continue reading

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a^2 + b^2 + c^2 = d^2

    Prove that the product of the positive integers a and b is even if and only if there exist positive integers c and d such that                         … Continue reading

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Smallest integer N; N = a+b+c

    (1) Find the smallest integer   such that     are distinct positive integers,   and (1)   ,    ,    and      are perfect squares. (2)   ,    ,    and      are … Continue reading

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When (a^2 + 2b^2) and (a + 2b) are squares with gcd(a,b) = 1

  Find nonzero integers     such that   and     are squares and     …………… ………….. ………… ………… ………… ………… ……….. ……….. ………. ….. It’s not the solution we are looking for. ………. …..   (a, b) … Continue reading

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(10*a + b) * c = a^2 + b^2 + c^2

      Paul found – Part 1   Pattern #2 :                    

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Integer n and the sum of the first n positive integers are both squares

    Paul found:              

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10*a + b – a^2 – b^2 = 10*x + y – x^2 – y^2

  Part 1 :   2-digit numbers   Find more solutions                                                      ——————————————   Part 2      

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Products| (1+2)(3+4+…+n) is a square, n ≤ 100

      Find all   n ≤ 100   so that (1 + 2 + 3 + 4)(5 + 6 + … + n) (1 + 2 + 3 + 4 + 5)(6 + 7 + … + n) … Continue reading

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