Quant Boosters - Soumya Chakraborty - Set 3



  • a^2 + ab + b^2 = n^2
    (a+b)^2 = n^2 + ab
    (a+b+n)(a+b-n) = ab

    as 'a' and 'b' are prime numbers
    ab has 4 factors: 1, a, b, ab
    obviously, a + b + n > a + b - n

    Case 1:
    a + b + n = ab
    a + b - n = 1

    Case 2:
    a + b + n = a
    a + b - n = b ... considering a>b

    But, case 2 is not possible.
    b = -n
    a = n,
    a = -b ... which is impossible

    Case 1:
    adding the two equations
    2a + 2b = ab + 1
    2a - ab + 2b = 1
    a(2-b) - 2(2-b) = -3
    (a-2)(b-2) = 3

    only possibility for 'a' and 'b' being prime and satisfying the above condition is: 5 and 3

    giving us he only possibility 49


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