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#1 2013-04-28 12:48:26

Agnishom
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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'
'Who are you to judge everything?' -Alokananda

#2 2013-04-28 13:24:16

bobbym
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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.
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.

#3 2013-04-28 13:57:18

Agnishom
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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'
'Who are you to judge everything?' -Alokananda

#4 2013-04-28 14:09:25

bobbym
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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.
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.

#5 2013-04-28 17:13:35

bob bundy
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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.   dizzy

Bob


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

#6 2013-04-28 19:01:34

bobbym
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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.
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.

#7 2013-04-28 19:27:42

Agnishom
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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'
'Who are you to judge everything?' -Alokananda

#8 2013-04-28 19:29:32

bobbym
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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.
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.

#9 2013-04-28 19:39:39

Agnishom
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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'
'Who are you to judge everything?' -Alokananda

#10 2013-04-28 19:48:35

bobbym
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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.
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.

#11 2013-04-29 04:00:37

Nehushtan
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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 (2013-04-29 04:00:57)


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#12 2013-05-17 20:39:02

Agnishom
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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'
'Who are you to judge everything?' -Alokananda

#13 2013-05-17 21:00:08

bobbym
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Re: Number Theory Problems

Hi;


In mathematics, you don't understand things. You just get used to them.
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.

#14 2013-05-17 21:33:20

Agnishom
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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'
'Who are you to judge everything?' -Alokananda

#15 2013-05-17 21:44:30

bobbym
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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.
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.

#16 2013-05-17 22:10:08

Agnishom
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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'
'Who are you to judge everything?' -Alokananda

#17 2013-05-18 00:34:43

Nehushtan
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Re: Number Theory Problems











It is easily checked that 677 is prime.


PS: In general:


In the above problem:

Last edited by Nehushtan (2013-05-18 01:34:36)


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#18 2013-05-18 03:24:49

Agnishom
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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'
'Who are you to judge everything?' -Alokananda

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