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Math Help - positive integer with 13 factors

  1. #1
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    positive integer with 13 factors

    What is the lowest positive integer with exactly 13 factors, both the number and 1 count as a factor.


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  2. #2
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    Hello, reagan3nc!


    What is the lowest positive integer with exactly 13 factors?
    Both the number and 1 count as factors.

    Most numbers have an even number of divisors.

    When factoring, the divisors appear in pairs.

    . . 24 \;=\;\begin{Bmatrix}1\cdot24 \\ 2 \cdot 12 \\ 3\cdot 8 \\ 4 \cdot 6 \end{Bmatrix}\;\;\hdots\text{ 8 factors}


    The only numbers with an odd number of factors are squares.

    . . 36 \:=\:\begin{Bmatrix}1\cdot36 \\ 2\cdot 18 \\ 3\cdot12 \\ 4\cdot9 \\ 6\cdot6\end{Bmatrix}\;\;\hdots\text{9 factors}


    The least integer with 13 divisors is: . 2^{12} \:=\:\boxed{4096}

    . . The divisors are: . 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096

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