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Math Help - Base Numbers

  1. #1
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    Base Numbers

    The number of possible bases b for which 12321(base b) is a perfect square is:
    a] 1
    b] 2
    c] 3
    d] 4
    e] more than 4
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  2. #2
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    Hello, foreverbrokenpromises!

    I found the answer by experimenting . . .


    The number of possible bases b for which 12321_b is a perfect square is:

    . . (a)\;1 \qquad (b)\;2 \qquad (c)\;3 \qquad (d)\;4 \qquad (e)\text{ more than 4}

    The number 12321_b is equal to: . b^4 + 2b^3 + 3b^2 + 2b + 1

    . . which is already a perfect square! . (b^2 + b + 1)^2


    Since the number uses {1, 2, 3}, the base must be at least 4.


    Therefore, 12321_b is a perfect square for any base b \ge 4 . . . answer (e)


    ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

    Examples:

    . . \begin{array}{cccccc}12321_4 &=& 441 &=& 21^2 \\ 12321_5 &=& 961 &=& 31^2 \\ 12321_6 &=& 1849 &=& 43^2 \\ 12321_9 &=& 8281 &=& 91^2 \\ 12321_{12} &=& 24,\!649 &=& 157^2 \end{array}

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