1. Not finding help here? Sign up for a free 30min 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!

Simple division proof

  1. Jan 3, 2012 #1
    1. The problem statement, all variables and given/known data


    Prove that for every integer n, 15|n iff 5|n and 3|n.

    I have proven the "if" direction. My question regards the "only if" portion of the proof.
    So far I have:

    (<--) Suppose 5|n and 3|n. Then there are j,k ∈ Z for which n = 5j and n = 3k.

    *above, Z stands for the set of integers

    I have found that both (j + k)/8 and (j - k)/2 seem to provide the desired value to show that 15|n, however I am not entirely sure how to show that either of these values is an integer (as is required by the definition of "divides"). I assume I am not proposing a clever enough value (or form) for the divisor. Any advice on this?


    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Jan 3, 2012 #2
    Heh heh, stupid me. It looks like a better form of the desired value is 2k - j, which of course an integer. The proof of the "only if" direction then goes:

    (<---) Suppose 3|n and 5|n. Then there are j,k ∈ Z for which n = 5j and n = 3k. Note, then, that (2j - k) ∈ Z and also:
    (15)(2j - k) = (30j - 15k) = (6)(5j) - (5)(3k) = 6n - 5n = n.

    =)
     
    Last edited: Jan 3, 2012
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Simple division proof
  1. Division Proof (Replies: 1)

Loading...