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Odd Integer and Multiple of Four

  1. Nov 2, 2010 #1
    1. The problem statement, all variables and given/known data

    Suppose that n is an odd integer. Prove that n is either one greater than a multiple of 4 or one less than a multiple of 4.

    2. Relevant equations


    3. The attempt at a solution

    I realize that this is going to be a direct proof. However, I am stumped on where to go from here.
  2. jcsd
  3. Nov 2, 2010 #2
    We know that the if n is a odd integer it is in form of 2a+1.

    Do two cases:
    Case 1 a is even so a = 2b for some b
    Case 2: a is odd so a = 2b+1 for some b (hint: express 3 in terms of 4)
    Last edited: Nov 2, 2010
  4. Nov 2, 2010 #3


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    A slightly different way: any integer must be of one of these forms:
    a) 4n
    b) 4n+ 1
    c) 4n+ 2
    d) 4n+ 3
    for some n. Both 4n= 2(2n) and 4n+2= 2(2n+1) are even. Can you show that 4n+ 3= 4m- 1 for some number m?
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