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## #1 2013-04-27 14:48:26

Agnishom
Real Member
From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,833
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### Number Theory Problems

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

'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'
I'm not crazy, my mother had me tested.

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## #2 2013-04-27 15:24:16

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### Re: Number Theory Problems

Hi;

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

answer = 1, 3, 15, 21, 25

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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## #3 2013-04-27 15:57:18

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,833
Website

### Re: Number Theory Problems

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'
I'm not crazy, my mother had me tested.

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## #4 2013-04-27 16:09:25

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### Re: Number Theory Problems

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.
Always satisfy the Prime Directive of getting the right answer above all else.

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## #5 2013-04-27 19:13:35

bob bundy
Registered: 2010-06-20
Posts: 8,082

### Re: Number Theory Problems

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

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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## #6 2013-04-27 21:01:34

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### Re: Number Theory Problems

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.
Always satisfy the Prime Directive of getting the right answer above all else.

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## #7 2013-04-27 21:27:42

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,833
Website

### Re: Number Theory Problems

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'
I'm not crazy, my mother had me tested.

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## #8 2013-04-27 21:29:32

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### Re: Number Theory Problems

bobbym wrote:

The answers are these numbers squared.

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

3^2 = 9

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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## #9 2013-04-27 21:39:39

Agnishom
Real Member
From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,833
Website

### Re: Number Theory Problems

Ooh, do we search them manually?

'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'
I'm not crazy, my mother had me tested.

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## #10 2013-04-27 21:48:35

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### Re: Number Theory Problems

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.

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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## #11 2013-04-28 06:00:37

Nehushtan
Member
Registered: 2013-03-09
Posts: 904
Website

### Re: Number Theory Problems

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.

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

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## #12 2013-05-16 22:39:02

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,833
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### Re: Number Theory Problems

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

'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'
I'm not crazy, my mother had me tested.

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## #13 2013-05-16 23:00:08

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### Re: Number Theory Problems

Hi;

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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## #14 2013-05-16 23:33:20

Agnishom
Real Member
From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,833
Website

### Re: Number Theory Problems

How did you come into that formula?

'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'
I'm not crazy, my mother had me tested.

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## #15 2013-05-16 23:44:30

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### Re: Number Theory Problems

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.

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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## #16 2013-05-17 00:10:08

Agnishom
Real Member
From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,833
Website

### Re: Number Theory Problems

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

'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'
I'm not crazy, my mother had me tested.

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## #17 2013-05-17 02:34:43

Nehushtan
Member
Registered: 2013-03-09
Posts: 904
Website

### Re: Number Theory Problems

It is easily checked that 677 is prime.

PS: In general:

In the above problem:

Last edited by Nehushtan (2013-05-17 03:34:36)

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## #18 2013-05-17 05:24:49

Agnishom
Real Member
From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,833
Website

### Re: Number Theory Problems

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?

'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'
I'm not crazy, my mother had me tested.

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