I am trying to prepare for my final exam in NT. Any help would be appreciated. These questions are just for study purposes, I messed them up on my homework, etc.

Reduced Residue System Proof

Show that if a=b(mod n) then (a,n)= (b,n)

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- Jul 27th 2009, 12:06 PMdiddledabbleHelp Preparing for Final Exam
I am trying to prepare for my final exam in NT. Any help would be appreciated. These questions are just for study purposes, I messed them up on my homework, etc.

Reduced Residue System Proof

Show that if a=b(mod n) then (a,n)= (b,n) - Jul 27th 2009, 04:54 PMGamma
(mod n) for some integer k

Now we show the set of divisors of a and n are the same as the set of divisors of b and n.

Let d divide a and n.

then d divides the RHS so it divides b as well.

Let d divide b and n.

then d divides the LHS so it divides a as well.

Thus the set of divisors is the same, so their greatest element is the same. so (a,n)=(b,n) - Aug 3rd 2009, 11:13 PMstreethot
First of all, assuming that we have that if and then .

Proof:

and

then

Now, let and . Then , so . Hence .

Similarly, , so . Hence . Therefore - Aug 4th 2009, 12:55 PMdiddledabbleAnother Proof
Show that if 2n+1 is prime, then n must be a power of 2.

- Aug 4th 2009, 01:01 PMGammaNot true
- Aug 4th 2009, 01:08 PMGamma
Maybe what you meant was being prime implies for . If so, I would check out this:Fermat number - Wikipedia, the free encyclopedia

For completeness, it should state the n could also be 0. - Aug 4th 2009, 01:09 PMdiddledabble
- Aug 4th 2009, 01:12 PMGamma
- Aug 4th 2009, 01:13 PMstreethot
Gamma,

Show that**IF**2n+1 is prime, then n must be a power of 2. - Aug 4th 2009, 01:16 PMdiddledabble
Streethot, Everytime I read it I think of a different way to go with it. I think you have it though. 2^n+1 must be a prime to start with.

- Aug 4th 2009, 01:17 PMGamma
- Aug 4th 2009, 01:23 PMdiddledabble
But if it should have been let n=2 that is not a square. I just don;t get this one and fermat seems way to tough to be on the exam.

- Aug 4th 2009, 01:30 PMstreethot
Sorry, i thought the question was about with

- Aug 4th 2009, 01:33 PMGamma
The statement should be

IF is prime, then n is a power of 2 or n=0.

the proof is given here Fermat number - Wikipedia, the free encyclopedia

There is nothing complicated about the proof supplied in the link. It is a proven fact, this has to be what the teacher was going for, the other possibilities are either not true or would never be asked on an exam by someone with a PhD in math.

I am not following what your problem is with this, it is a very straightforward if then statement. The proof goes by contradiction in supposing that n is not a power of two, then it has an odd prime factor. Then you show how you factor which makes it NOT prime, a contradiction. - Aug 4th 2009, 01:35 PMGamma