Results 1 to 5 of 5
Like Tree1Thanks
  • 1 Post By richard1234

Math Help - Prove that 5 divides at least one of {m,n,p}.

  1. #1
    Junior Member
    Joined
    Apr 2012
    From
    Alaska
    Posts
    31

    Prove that 5 divides at least one of {m,n,p}.

    For m,n,p belong to Z, suppose that 5 divides m^2+n^2+p^2. Prove that 5 divides at least one of {m,n,p}. (i.e. Prove if m^2+n^2+p^2 is congruent to 0 mod 5, then m is congruent to 0 mod 5, n is congruent to 0 mod 5 or p is congruent to 0 mod 5.)
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Oct 2009
    Posts
    5,540
    Thanks
    780

    Re: Prove that 5 divides at least one of {m,n,p}.

    Find all possible nonzero squares modulo 5 and see if three of them can add up to 0.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Nov 2011
    Posts
    17

    Re: Prove that 5 divides at least one of {m,n,p}.

    Quote Originally Posted by emakarov View Post
    Find all possible nonzero squares modulo 5 and see if three of them can add up to 0.
    I don't really get how this would help with the proof.

    I did like this:

    Assume none of {m,n,p} is divisible by 5. This implies that none of {m^2,n^2,p^2} is divisible by 5, i.e.
    m^2 ≢ 0 mod 5
    n^2
    ≢ 0 mod 5
    p^2
    ≢ 0 mod 5
    m^2 + n^2 + p^2
    ≢ 0 mod 5 which leads to a contradiction => one of {m,n,p} must be divisible by 5.

    Is this a valid proof?
    Last edited by MagisterMan; July 27th 2012 at 08:16 AM.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Joined
    Oct 2009
    Posts
    5,540
    Thanks
    780

    Re: Prove that 5 divides at least one of {m,n,p}.

    Quote Originally Posted by MagisterMan View Post
    Assume none of {m,n,p} is divisible by 5. This implies that none of {m^2,n^2,p^2} is divisible by 5, i.e.
    m^2 ≢ 0 mod 5
    n^2
    ≢ 0 mod 5
    p^2
    ≢ 0 mod 5
    m^2 + n^2 + p^2
    [COLOR=#333333][FONT=arial]≢ 0 mod 5
    My browser does not show any symbol between "m^2" and "0 mod 5". I assume you mean m^2 is not congruent to 0 modulo 5. It is not clear why you conclude that m^2 + n^2 + p^2 is not congruent to 0. For example, if m^2 = 3 and n^2 = p^2 = 1 mod 5, then m^2 + n^2 + p^2 = 0 mod 5. The problem is that m^2 can't be 3 mod 5.

    I also suggested proving this fact by contradiction and finding all possible values of m^2 mod 5. Then see if three such values can add up to zero.
    Follow Math Help Forum on Facebook and Google+

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

    Re: Prove that 5 divides at least one of {m,n,p}.

    Quote Originally Posted by MagisterMan View Post
    I don't really get how this would help with the proof.

    I did like this:

    Assume none of {m,n,p} is divisible by 5. This implies that none of {m^2,n^2,p^2} is divisible by 5, i.e.
    m^2 ≢ 0 mod 5
    n^2
    ≢ 0 mod 5
    p^2
    ≢ 0 mod 5
    m^2 + n^2 + p^2
    ≢ 0 mod 5 which leads to a contradiction => one of {m,n,p} must be divisible by 5.

    Is this a valid proof?
    No. Maybe m^2 \equiv n^2 \equiv 2 and p^2 \equiv 1 (mod 5).

    Suppose m,n,p are not 0 (mod 5). Modulo 5

    1^2 \equiv 1
    2^2 \equiv 4
    3^2 \equiv 4
    4^2 \equiv 1

    It is impossible to have three such squares add up to 0 (mod 5). Therefore at least one of m,n,p is congruent to 0 (mod 5).
    Thanks from Deveno
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Prove using induction on n that 3 divides (4^n)+5
    Posted in the Number Theory Forum
    Replies: 16
    Last Post: July 2nd 2011, 06:38 AM
  2. Prove 3 divides two integers
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: September 9th 2010, 09:38 AM
  3. Prove that ab divides c.
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: March 3rd 2010, 09:42 AM
  4. Prove that gcd(a,b) divides lcm[a,b]
    Posted in the Number Theory Forum
    Replies: 5
    Last Post: February 15th 2010, 07:44 PM
  5. prove by induction 6 divides n^3 -n
    Posted in the Discrete Math Forum
    Replies: 7
    Last Post: June 1st 2008, 04:08 AM

Search Tags


/mathhelpforum @mathhelpforum