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Homework Help: 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
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