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Math Help - 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: . N \:=\:p^a\cdot q^b\cdot r^c\,\cdots .
    (prime factorization)

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

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


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

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


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

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

    . . And the smallest K occurs when 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.

    4096=2^{12}

    4096=1.2.2.2.2.2.2.2.2.2.2.2.2
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