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Thread: Determining if n is a square from number of divisors?

  1. #1
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    Determining if n is a square from number of divisors?

    I've noticed for a few cases that if you compute all the divisors of a composite number, if this list of divisors (including 1 and the number) is even, then the number is not a square, and if it's odd, it is a square.

    But that's just for some specific examples like

    divisors of 30 : {1, 2, 3, 5, 6, 10, 15, 30 } : even # of divisors, and not a square

    divisors of 36: {1, 2, 3, 4, 6, 9, 12, 18, 36}: odd # of divisors, and is a square.

    I'm wondering is this just a coincidence, or does this hold for all numbers? And are there any correlations between number of divisors and if the number is a cube, a quadruple, etc?
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  2. #2
    MHF Contributor Bruno J.'s Avatar
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    Good eye! Your observation is correct. This is because, if $\displaystyle n=p_1^{\alpha_1}...p_m^{\alpha_m}$ then the number of divisors of $\displaystyle n$ is $\displaystyle (\alpha_1+1)...(\alpha_m+1)$. (Can you prove it?) In the case of a square all $\displaystyle \alpha$'s are even.

    You can probably answer your other question with the help of the formula above.
    Last edited by Bruno J.; Aug 21st 2009 at 07:39 AM.
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by Bruno J. View Post
    Good eye! Your observation is correct. This is because, if $\displaystyle n=p_1^{\alpha_1}...p_m^{\alpha_m}$ then the number of divisors of $\displaystyle n$ is $\displaystyle (\alpha_1+1)...(\alpha_m+1)$. (Can you prove it?) In the case of a square all $\displaystyle \alpha$'s are odd.

    You can probably answer your other question with the help of the formula above.
    In the case of a square all $\displaystyle \alpha$'s are even.

    CB
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  4. #4
    Senior Member TheAbstractionist's Avatar
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    Quote Originally Posted by QM deFuturo View Post
    I've noticed for a few cases that if you compute all the divisors of a composite number, if this list of divisors (including 1 and the number) is even, then the number is not a square, and if it's odd, it is a square.

    But that's just for some specific examples like

    divisors of 30 : {1, 2, 3, 5, 6, 10, 15, 30 } : even # of divisors, and not a square

    divisors of 36: {1, 2, 3, 4, 6, 9, 12, 18, 36}: odd # of divisors, and is a square.

    I'm wondering is this just a coincidence, or does this hold for all numbers? And are there any correlations between number of divisors and if the number is a cube, a quadruple, etc?
    Note that if $\displaystyle d$ is a divisor of $\displaystyle n$ then so is $\displaystyle \frac nd.$ Hence, all the divisors of $\displaystyle n$ occur in pairs $\displaystyle \left\{d,\,\frac nd\right\}.$ Furthermore $\displaystyle d=\frac nd\ \iff\ n=d^2$ is a perfect square. It folllows that if $\displaystyle n$ is not a perfect square, then it has an even number of divisors (since each pair of divisors $\displaystyle d,\,\frac nd$ will be distinct), while if $\displaystyle n$ is a perfect square then it has an odd number of divisors (the pairs $\displaystyle d,\,\frac nd$ are all distinct except for the divisor $\displaystyle d=\sqrt n\,).$
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  5. #5
    MHF Contributor Bruno J.'s Avatar
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    Quote Originally Posted by CaptainBlack View Post
    In the case of a square all $\displaystyle \alpha$'s are even.

    CB
    Haha. I can't believe I said that.
    Obviously I meant "all the $\displaystyle \alpha+1$".
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