# Finding the number of odd integers $0 < n < 1000$ such that its number of divisors divides $n$

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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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

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 think they are 5 : 1,9,225,441,625

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