1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Congruency and Divisibility of Odd in Z

  1. Feb 27, 2014 #1
    1. The problem statement, all variables and given/known data
    Let n be any odd integer. Prove that 1 is the only "common" divisor of the integers n and n+2.

    3. The attempt at a solution
    I don't think I understand the question.

    The few notes I have state d| (n+2 )- n

    This resembles n+2 ##\equiv## n mod d , but I don't see the connection.
    The congruency means that n+2 and n share the same remainder.
    Any help?
     
  2. jcsd
  3. Feb 27, 2014 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    The problem doesn't have anything to do with congruency. If d is a common divisor then d divides n and d divides n+2, so it follows that d must divide (n+2)-n. Don't you agree? Now use that to solve the problem.
     
  4. Feb 27, 2014 #3
    Thank you. I agree.
    I understand that
    ##d | n+2 -n ##

    ## n+2 -n = dq, ## where q ## \in Z##
    Since, 2 = dq, so d and q could be 1 or 2.
     
  5. Feb 27, 2014 #4

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Correct, that d must be 1 or 2. Now can you prove the original problem?
     
  6. Feb 27, 2014 #5
    I thought that I already did.
    d |n and d| n+2
    n =dk and n+2 = dp, k and p are in Z


    n is odd, so n =2a+1 for some a in Z.

    dk +2 =dp
    2a+1 +2 = dp
    2(a+1) + 1 = dp

    I think this means that I am confused.
     
  7. Feb 27, 2014 #6

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Yeah, it does look like you have lost your way. It's much simpler than that. You know d=1 or 2. You are given that n is odd. d divides n. Can d be 2?
     
  8. Feb 27, 2014 #7
    An even number can never divide an odd number. 'd' must be 1. :>

    Thanks.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Congruency and Divisibility of Odd in Z
  1. Linear congruence (Replies: 4)

  2. Congruence problem. (Replies: 10)

  3. Linear congruence (Replies: 3)

Loading...