#### Agnishom wrote:

1.For how many odd positive integers n<1000 does the number of positive divisors of n divide n?

As **bobbym** pointed out, *n* must be a perfect square. *n*=1 is one possibility. For the others, it can be easily checked that all odd perfect squares greater than 1 and less than 1000 are have at most two distinct prime factors in their factorization. Thus the possibilities for *n*>1 are:

where

*p* and

*q* are distinct primes and

*a*,

*b* positive integers.

First case:

The number of positive divisors of

*n* are

– i.e. there are

positive divisors. So the possibilites are

and

. (Not

; that would make

*n* too large.)

Second case:

There are only two such

possible, namely

and

. The number of positive divisors for each number is 9, which does divide each number.

Therefore the answer to your question is:

**There are 5 odd numbers less than 1000 which are divisible by their number of positive divisors**, namely 1, 9, 225, 441, and 625.