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How to solve this: For how many odd positive integers $n<1000$ does the number of positive divisors of $n$ divide $n$?

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3 Answers

Hint: Notice that every number has an even number of divisors except the square numbers, since factors occur in pairs. Odd numbers must have only odd factors.

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I found the following numbers 9, 225, 441, & 625. 4 should be the answer

Number of Divisors of number expressed of the form $a_1^{b_1}\cdot a_2^{b_2}\cdot a_3^{b_3}\cdots a_r^{b_r}$

where $a_1$, $a_2$ are primes and $b_1$, $b_2$ are their respective powers..

then no of divisors is $1\cdot(b_1+1)\cdot(b_2+1)\cdot(b_3+1)\cdots(b_r+1)$

and therefore Only square numbers can have an Odd number of divisors

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Also 1.${}{}{}{}$ –  Kundor Apr 24 '13 at 16:25
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I think they are 5 : 1,9,225,441,625

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While correct, your answer provides no insight as to how the problem is solved, nor any more information than the previous answers. You might get more votes if you explain how you got your answer. –  robjohn Apr 27 '13 at 16:19
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