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Complicated divisibility problem

  1. May 25, 2010 #1
    1. The problem statement, all variables and given/known data

    If 5 divides m^2 + n^2 + p^2 , prove that 5 divides wither m, or n, or p.

    2. Relevant equations

    m,n,p are all integers

    3. The attempt at a solution

    I am having some major problems with this chapter on modular arithmetic. any help is much appreciated!!!

    modular arithmetic is not needed to solve the problem, but may be helpful.
     
    Last edited: May 25, 2010
  2. jcsd
  3. May 25, 2010 #2
    What possible values can m^2 have mod 5?
     
  4. May 25, 2010 #3
    Well if we are considering divisibility by 5, i can let
    m=5k+r, where k is some integer and r may be 0, 1, 2, 3, or 4.

    So m^2 = 25k^2 + 10kr +r^2
    so
    m^2 ≡ n mod 5 equals:

    25k^2 + 10kr +r^2 ≡ n mod 5
    and since 25k^2 + 10kr ≡ 0 mod 5, we are left with
    r^2 ≡ n mod 5

    so n may be 1, 4, or 0. Am I going in the right direction? Thanks for the reply.
     
  5. May 25, 2010 #4
    That's correct. So what can m^2 + n^2 + p^2 be if none are congruent to 0 mod 5?
     
  6. May 25, 2010 #5
    thanks for your help.
     
    Last edited: May 25, 2010
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