# Homework Help: Simple division proof

1. Jan 3, 2012

### Syrus

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. Jan 3, 2012

### Syrus

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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