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

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
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Hi;

Hint: Only squares can have an odd number of divisors, that limits the search to 16 numbers.

answer = 1, 3, 15, 21, 25

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Now, how can you say that those number's divisors have to be odd?

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

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**bobbym****Administrator**- From: Bumpkinland
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I am not sure what you are exactly asking so I will answer every possible question.

There is a formula to compute the number of positive divisors of any integer.

those number's divisors have to be odd

Odd numbers have odd divisors.

Even numbers must have one 2 in there prime factorization at least.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.** **A number by itself is useful, but it is far more useful to know how accurate or certain that number is.**

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**bob bundy****Moderator**- Registered: 2010-06-20
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Agnishom wrote:

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

I'm not following this thread at all.

Let's take n = 3

divisors are {1,3} so the number of them is 2.

2 does not divide 3

Take n = 9

divisors are {1,3,9} That's 3 divisors. 3 divides 9.

I must be misunderstanding something, but I don't know what.

Bob

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
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Hi Bob;

The answers are these numbers squared.

1, 3, 15, 21, 25 as given above.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.** **A number by itself is useful, but it is far more useful to know how accurate or certain that number is.**

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And why not 9 as bob told?

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
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bobbym wrote:

The answers are these numbers squared.

1, 3, 15, 21, 25 as given above.

3^2 = 9

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Ooh, do we search them manually?

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
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That is how I did it. You just square 1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31 and check.

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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

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.

*Last edited by Nehushtan (2013-04-28 06:00:57)*

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What is the largest prime factor of 5^8 + 2^2?

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
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Hi;

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**

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How did you come into that formula?

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
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There are things called aurifeuillian factorizations.

This one could be the basis for many others. But like Aurifeuille who used it for n = 14 in 1871 there is much trial and error.

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Isn't it just the a^2 - b^2 formula?

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Oh Good one! Thanks!

It is easily checked that 677 is prime.

By trying all of 2,3,5,7,11,13,17,19, and 23?

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