How to Prove Product of Divisors Formula and Derive Largest Unattainable Score?

[logic2b1]

1)Prove : the product of all of the positive divisors of n ( including n itself ) is
n^(d(n)/2).

2) Suppose you have a game in which there are two kinds of scoring events. One event gives a score of m points, and the other gives a score of n points. Asusume that m and n are relatively prime, and derive a formula for the largest unattainable score. Prove your answer is correct.

 

[shmoe]

Hi, you'll find you get more help if you post what you have tried, so we can see where you are stuck and advise from there. So just a couple of hints for now:

1)Prove : the product of all of the positive divisors of n ( including n itself ) is
n^(d(n)/2).

You might find it easier to break this into two cases, n a perfect square, and n not a perfect square.

2) Suppose you have a game in which there are two kinds of scoring events. One event gives a score of m points, and the other gives a score of n points. Asusume that m and n are relatively prime, and derive a formula for the largest unattainable score. Prove your answer is correct.

I can't think of any good hints that don't give away too much here. Have you tried working out some examples and attempting to guess a formula? The formula will be fairly simple in terms of m and n, so this shouldn't be a hopeless way to start.

 

[Office_Shredder]

I want to try the first one (I'm doing the second one right now)... what is d(n)?

 

[shmoe]

d(n)=the number of divisors of n

 

[benorin]

Start with the prime factorization of n,

$$n=p_{1}^{\alpha_{1}}p_{2}^{\alpha_{2}}\cdots p_{r}^{\alpha_{r}}$$

where the $$p_{i}'s$$ are primes. Now use numbers of that divide the above to form the product. You can find some info on the function d(n) here.

 

[logic2b1]

thks for your help guys!

Sorry, I am on a short vocation and is not so convenience to log into the internet. I will be back home two days later.

If you guys have more idea, please do advise me. Thank you very much for your help.

 

[neurocomp2003]

logic look up two concepts in number theory ...Euler Phi Function and Euler Sigma Function(this latter may just becalled Euler Sigma, or Sigma Function)
It'll tell you how to find the product of all divisors

 

[Gokul43201]

Balakrishnan, (to address the reason for the deleted post) your participation in these forums is much appreciated. However, the policy here is to absolutely not help with what are homework/coursework problems UNLESS the OP shows at least some sign of having made original effort. And posting complete solutions, as you have, do more to harm than help the OP - at least as far as this site is concerned. This is specified clearly in the Posting Guidelines.

Logic: Please read the posting guidelines, for standard textbook problems. You'll find a link in my signature.

Also, this thread belongs in the Homework Help section, not here. Please make sure you post to the correct location next time.