# Discrete Math, necessary versus sufficient condition question.

I was given this question, and I'm a little confused. A little bit of help would be great.

Find a condition on $m$ that is sufficient but not necessary for $\frac{m}{2} \in \mathbb{Z}$.

I get that if a condition $x$ is sufficient for $y$, the presence of $x$ guarantees the presence of $y$. Applying that to this question, am I looking for something like a condition $x$ that guarantees the validity of $\frac{m}{2} \in \mathbb{Z}$, but $\frac{m}{2} \in \mathbb{Z}$ is possible without the presence of $x$?

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A pedantically-correct answer to your question would be "yes" :P if you want more than that, maybe ask for more? – Ben Millwood Sep 17 '12 at 23:26

Yes, you are right. You are seeking a condition, given which the statement holds, but without which the statement could still hold. In a sense, you have to find a condition that asks too much.

That said, $\dfrac m2\in\mathbb{Z}$ means exactly that $m$ is an even number, i.e. a multiple of 2: $m$ being an even number is a necessary and sufficient condition (it's the $\Leftrightarrow$ symbol, each thing implies the other).
You need something that is not necessary, so this can't be the answer yet. Since it is something sufficient, though, you are on the right track: now try asking more!

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1/4m? I get that, thanks so much. But I'm trying to think of one that is now necessary but not sufficient. I'm totally unable to grasp this one. – Doug Smith Sep 18 '12 at 1:26
Well, in this case you have to ask too little w.r.t. the necessary and sufficient condition. For example, you could ask $m\in\mathbb{Z}$, or $m\in\mathbb{Q}$. – Andrea Orta Sep 18 '12 at 6:27
Dear Doug, if you are satisfied with this answer, please accept it. That way, your question won't be seen as unanswered anymore (see here). Obviously, if you still have any doubts, feel free to ask! – Andrea Orta Sep 22 '12 at 11:01

$m/4\in \mathbb Z$

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I don't want an answer, I want an explanation. >_< – Doug Smith Sep 17 '12 at 22:57
@DougSmith While your question does ask for "a little bit of help", it would be best to specify explicitly in your question that you don't want a full answer. – Quinn Culver Sep 21 '12 at 0:36