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Homework Help: Consecutive odd natural numbers - one is composite. Prove!

  1. Sep 29, 2010 #1
    1. The problem statement, all variables and given/known data

    Every triple of consecutive odd natural numbers, with the first being at least 5, contains at least on composite.

    2. Relevant equations

    N/A

    3. The attempt at a solution

    I know from number theory that of every set of consecutive odd integers, one of them is divisible by three, thereby making it a composite number. I just don't know how to prove it. Can anybody help?
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Sep 29, 2010 #2

    HallsofIvy

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    Science Advisor

    A triple of consectutive odd integers can be written as 2n+1, 2n+ 3, and 2n+ 5.

    Suppose 2n+ 1 is NOT a multiple of 3. Then it has a remainder of either 1 or 2 when divided by 3

    a) Suppose 2n+ 1 has remainder 1 when divided by 3: 2n+ 1= 3k+ 1 so that 2n= 3k. Look at both 2n+3 and 2n+ 5 in this case.

    b) Suppose 2n+ 1 has remainder 2 when divided by 3: 2n+ 1= 3k+ 2 so that 2n= 3k+ 1. Look at both 2n+3 and 2n+ 5 in this case.
     
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