Results 1 to 6 of 6
Like Tree1Thanks
  • 1 Post By richard1234

Math Help - Number Theory 1

  1. #1
    Newbie
    Joined
    Jul 2012
    From
    Delhi
    Posts
    12

    Number Theory 1

    I add first N natural numbers and finds the sum to be 1850 . But actually one number was added twice by mistake .
    Find the difference between N and that number ?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Prove It's Avatar
    Joined
    Aug 2008
    Posts
    10,964
    Thanks
    1008

    Re: Number Theory 1

    Quote Originally Posted by timkuc View Post
    I add first N natural numbers and finds the sum to be 1850 . But actually one number was added twice by mistake .
    Find the difference between N and that number ?
    The sum of the first N natural numbers is equal to \displaystyle \begin{align*} \frac{N(N+1)}{2} \end{align*}.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member
    Joined
    Jun 2012
    From
    AZ
    Posts
    616
    Thanks
    97

    Re: Number Theory 1

    1 + 2 + \dots + N + k = 1850, where k is the duplicated number.

    \frac{N(N+1)}{2} + k = 1850.

    Note that if N = 60, 60*61/2 = 1830. This means that k = 20, and the difference between N and k is 40.
    Thanks from timkuc
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Newbie
    Joined
    Jul 2012
    From
    Delhi
    Posts
    12

    Re: Number Theory 1

    so we have to do it intuitively (means by the guessing ) ? We can't get value by calculation right ?


    hmmm got it . thanks
    Last edited by timkuc; July 20th 2012 at 10:35 PM.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Super Member
    Joined
    Jun 2012
    From
    AZ
    Posts
    616
    Thanks
    97

    Re: Number Theory 1

    Yeah, guess-and-check seems to be the best solution.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor

    Joined
    Mar 2011
    From
    Tejas
    Posts
    3,150
    Thanks
    591

    Re: Number Theory 1

    well we know that N(N+1)/2 < 1850 so that:

    N(N+1) < 3700.

    that is: N2 + N - 3700 < 0

    using the quadratic equation, we have:

    N2 + N - 3700 = (N - (1/2)(19√41 - 1))(N + (1/2)(19√41 + 1)).

    for this to be < 0, the factors must be of different signs, and since N is assumed positive,

    we must have N - (1/2)(19√41 - 1) < 0, that is N < (1/2)(19√41 - 1) ~ 60.33

    so 60 is the largest natural number N could be.

    how small could N be? since k (the repeated number) must be ≤ N, we have:

    1850 ≤ N(N+1)/2 + N, so

    0 ≤ N2 + 3N - 3700, and thus:

    0 ≤ (N - (1/2)(√14809 - 3))(N + (1/2)(√14809 + 3)), which means N ≥ (1/2)(√14809 - 3) ~ 59.35

    this means that 60 is the smallest natural number N can be.

    therefore we conclude N = 60, whereupon the rest follows by richard1234's post.

    no guessing required.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Using group theory to prove results in number theory
    Posted in the Math Philosophy Forum
    Replies: 6
    Last Post: May 12th 2012, 01:34 AM
  2. Textbooks on Galois Theory and Algebraic Number Theory
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: July 8th 2011, 06:09 PM
  3. Replies: 2
    Last Post: December 18th 2008, 05:28 PM
  4. number theory
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: September 26th 2006, 06:58 PM
  5. Number theory, prime number
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: September 17th 2006, 08:11 PM

Search Tags


/mathhelpforum @mathhelpforum