Results 1 to 2 of 2

Math Help - This function is the so called set of divisors function...

  1. #1
    Junior Member
    Joined
    Nov 2011
    Posts
    26

    This function is the so called set of divisors function...

    Question Details
    We introduce the number of divisors function d previously. For this function, d: N->N, where d(n) is the number of natural number divisors of n.

    A function that is related to this function is the so-called set of divisors function. This can be defined as a function S that associates with each natural number the set of its distinct natural number factors. For example, S(6)={1,2,3,6}.

    1.) Does there exist a natural number n such that the absolute value of S(n) = 1? Explain.

    2.) Does there exist a natural number n such that the absolute value of S(n) =2? Explain?

    3.) Write the output for the function d in terms of the output for the function S. That is, write d(n) in terms of S(n).

    4.) Is the following statement true or false? Justify your conclusion.
    For all natural numbers m and n, if m doesn't equal n, then S(m) = S(n). If it's true, the justification should be a proof...

    5.) Is the following statement true or false? Justify your conclusion.
    For all sets T that are subsets of N, there exists a natural number n such that S(n) = T. Again, if it's true the justification should be a proof...

    Any help on this would be appreciated. This is a nongraded homework problem purely for my understanding and I have almost no idea how do do it, but I'll include my work below...


    1.) I said that there does exist a natural number such that S(n)=1, it is 1 itself, I think, but I'm not sure what that means? 1 is in a set by itself because it's its only divisor?

    2.) Along the same lines as part 1, would this S(n)=2,3,5,7,9,11, etc and all the other prime numbers, or is it just the set 2 by itself?

    3.) All I got is d(n)=S(n) but I know there is more to it than that.

    4.) and 5.) no idea
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Also sprach Zarathustra's Avatar
    Joined
    Dec 2009
    From
    Russia
    Posts
    1,506
    Thanks
    1

    Re: This function is the so called set of divisors function...

    Quote Originally Posted by Brjakewa View Post
    Question Details
    We introduce the number of divisors function d previously. For this function, d: N->N, where d(n) is the number of natural number divisors of n.

    A function that is related to this function is the so-called set of divisors function. This can be defined as a function S that associates with each natural number the set of its distinct natural number factors. For example, S(6)={1,2,3,6}.

    1.) Does there exist a natural number n such that the absolute value of S(n) = 1? Explain.

    2.) Does there exist a natural number n such that the absolute value of S(n) =2? Explain?

    3.) Write the output for the function d in terms of the output for the function S. That is, write d(n) in terms of S(n).

    4.) Is the following statement true or false? Justify your conclusion.
    For all natural numbers m and n, if m doesn't equal n, then S(m) = S(n). If it's true, the justification should be a proof...

    5.) Is the following statement true or false? Justify your conclusion.
    For all sets T that are subsets of N, there exists a natural number n such that S(n) = T. Again, if it's true the justification should be a proof...

    Any help on this would be appreciated. This is a nongraded homework problem purely for my understanding and I have almost no idea how do do it, but I'll include my work below...


    1.) I said that there does exist a natural number such that S(n)=1, it is 1 itself, I think, but I'm not sure what that means? 1 is in a set by itself because it's its only divisor?

    2.) Along the same lines as part 1, would this S(n)=2,3,5,7,9,11, etc and all the other prime numbers, or is it just the set 2 by itself?

    3.) All I got is d(n)=S(n) but I know there is more to it than that.

    4.) and 5.) no idea

    I think, you will find all the answer here:

    Divisor function - Wikipedia, the free encyclopedia
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Sum of divisors function question
    Posted in the Number Theory Forum
    Replies: 5
    Last Post: August 31st 2010, 06:04 AM
  2. Sum of positive divisors function
    Posted in the Number Theory Forum
    Replies: 4
    Last Post: March 17th 2010, 06:42 AM
  3. Sum of Positive Divisors Function
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: October 12th 2008, 03:36 PM
  4. The sum of postive divisors Function
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: August 4th 2008, 11:27 AM
  5. The sum of postive divisors Function
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: August 4th 2008, 11:24 AM

Search Tags


/mathhelpforum @mathhelpforum