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Thread: 4096 has 13 factors WHY

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    4096 has 13 factors WHY

    Why is 4096 the smallest number with 13 factors. What does 2^n have to do with it? I understand in order to get an odd number of factors you have to use a perfect square.


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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by reagan3nc View Post
    Why is 4096 the smallest number with 13 factors. What does 2^n have to do with it? I understand in order to get an odd number of factors you have to use a perfect square.


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    this seems like a follow up to this thread. you should have asked this question there

    powers of 2 would yield the smallest number. since powers of 1 would only yield 1 and thus would not reach 4096, and powers of 3 (or more, or a product with numbers greater than 2) would be greater than corresponding powers of 2 and so would not be minimum
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    Hello, reagan3nc!

    Why is 4096 the smallest number with 13 factors?
    What does 2^n have to do with it?
    I understand in order to get an odd number of factors you have to use a perfect square.
    There is a theorem that might help explain it.


    Given: .$\displaystyle N \:=\:p^a\cdot q^b\cdot r^c\,\cdots$ .
    (prime factorization)

    . . the number of factors of $\displaystyle N$ is: .$\displaystyle d(N) \:=\:(a+1)(b+1)(c+1)\cdots$

    . . . .
    Add one to each exponent and multiply.


    Example: .$\displaystyle N \:=\:6125 \:=\:5^3\cdot7^2$

    . . . $\displaystyle d(6125) \:=\:(3+1)(2+1) \:=\:12$ factors.


    We want a number $\displaystyle K$ so that: .$\displaystyle d(K) \:=\: 13 \:=\:12 + 1$

    Hence, $\displaystyle K$ has a prime factorization with an exponent of 12: .$\displaystyle K \:=\:p^{12}$

    . . And the smallest $\displaystyle K$ occurs when $\displaystyle p = 2.$


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    Thanks for all your help
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    Quote Originally Posted by reagan3nc View Post
    Why is 4096 the smallest number with 13 factors. What does 2^n have to do with it? I understand in order to get an odd number of factors you have to use a perfect square.


    Thanks for anything you can add to the picture.
    Since any power of 1 will give only 1 as answer. After this number 2 comes.

    $\displaystyle 4096=2^{12}$

    $\displaystyle 4096=1.2.2.2.2.2.2.2.2.2.2.2.2$
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