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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

I have the result, but I do not yet know how to get it.

All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

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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'

'Humanity is still kept intact. It remains within.' -Alokananda

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 84,183

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.

I have the result, but I do not yet know how to get it.

All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,291

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

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 84,183

Hi Bob;

The answers are these numbers squared.

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

I have the result, but I do not yet know how to get it.

All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

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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'

'Humanity is still kept intact. It remains within.' -Alokananda

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 84,183

bobbym wrote:

The answers are these numbers squared.

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

3^2 = 9

I have the result, but I do not yet know how to get it.

All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

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Ooh, do we search them manually?

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

'Humanity is still kept intact. It remains within.' -Alokananda

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 84,183

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.

I have the result, but I do not yet know how to get it.

All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

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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?

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

'Humanity is still kept intact. It remains within.' -Alokananda

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 84,183

Hi;

I have the result, but I do not yet know how to get it.

All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

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How did you come into that formula?

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

'Humanity is still kept intact. It remains within.' -Alokananda

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 84,183

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.

I have the result, but I do not yet know how to get it.

All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

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

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

'Humanity is still kept intact. It remains within.' -Alokananda

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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?

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

'Humanity is still kept intact. It remains within.' -Alokananda

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