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Math

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2 teams played each other recently. Team 1 scored N points, while team 2 scored M. 1<M and N<100.
The first digit of N is a, the second digit b. N-a+b^2.
The first digit of M is b, the second is a. By how many points does team M win the game? There is a mathematical way to do this, although I reached the answer by process of elimination.

  • Math -

    I'm sorry, N=a+b^2

  • Math -

    Let N = 10a + b
    but N = a + b^2
    so b^2 + a - 10a + b
    b^2 - b - 9a = 0
    b(b-1) = 9a

    which tell me that the product of 2 consecutive numbers, which are less than 10, must be a multiple of 9
    only possibilities
    9x8 = 72 yes
    8x7 = 56 no
    7x6 = 42 no
    6x5 = 30 no
    5x4 = 20 no
    4x3 = 12 no
    3x2 = 6 no
    2x1 = 2 no

    so 9a = 72
    a = 8
    then b^ - b - 72 = 0
    (b-9)(b+8) = 0
    b = 9

    N = a + b^2 = 8 + 81 = 89
    M = 98

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