Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    N/A

    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

    HallsofIvy

    User Avatar
    Science Advisor

    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?
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook