Results 1 to 3 of 3

Math Help - Prime numbers

  1. #1
    Newbie
    Joined
    Nov 2007
    Posts
    20

    Prime numbers

    Let n \ge 2 be a number that is not a prime.
    Show that there is a divisor p of n, with p \ge 2, so that n \ge p^2



    On my paper, I've written down n=rs where n and s are any natural numbers >1. Not sure if that's needed, though.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by Joel24 View Post
    Let n \ge 2 be a number that is not a prime.
    Show that there is a divisor p of n, with p \ge 2, so that n \ge p^2



    On my paper, I've written down n=rs where n and s are any natural numbers >1. Not sure if that's needed, though.
    If n>2 is not prime, then it has a factor p>=2, so there also exists a q>=2 such that pq=n. Now if both n<p^2 and n<q^2, then pq<n a contradiction.

    Hence every composite number >2 has a factor whose square is less than or equal to the number.

    RonL
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Nov 2007
    Posts
    20
    Ah, a proof (or sorts) by contradiction. I would've never have thought of that.

    Excellent. Thank you so much!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: October 22nd 2011, 01:37 PM
  2. Prime numbers
    Posted in the Algebra Forum
    Replies: 5
    Last Post: December 29th 2010, 05:54 AM
  3. prime numbers
    Posted in the Number Theory Forum
    Replies: 6
    Last Post: February 15th 2009, 09:34 AM
  4. which numbers are prime
    Posted in the Algebra Forum
    Replies: 2
    Last Post: March 9th 2008, 08:51 PM
  5. prime numbers
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: February 14th 2007, 08:21 PM

Search Tags


/mathhelpforum @mathhelpforum