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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
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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?

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'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.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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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
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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.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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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
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bobbym wrote:

The answers are these numbers squared.

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

3^2 = 9

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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Ooh, do we search them manually?

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

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**bobbym****Administrator**- From: Bumpkinland
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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.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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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
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Hi;

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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How did you come into that formula?

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**bobbym****Administrator**- From: Bumpkinland
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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.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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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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