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#1 2013-04-27 14:48:26
Number Theory Problems
1.For how many odd positive integers n<1000 does the number of positive divisors of n divide n?
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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
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
- Administrator
- Registered: 2010-06-20
- Posts: 10,645
Re: Number Theory Problems
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
Sometimes I deliberately make mistakes, just to test you! …………….Bob 
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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
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
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
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: 957
Re: Number Theory Problems
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
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
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
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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#18 2013-05-17 05:24:49
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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