How to find the values of $k$ if $2k \in \mathbb{Z}$ and $\frac{2}{k}\in \mathbb{Z}$

Question

Let $k \in \mathbb{R}$. What are the possible values of $k$ Under the condition that $2k \in \mathbb{Z}$ and $\frac{2}{k} \in \mathbb{Z}$. I know that the answer is $k \in \lbrace \frac{1}{2}, 1, 2 , -\frac{1}{2}, -1, -2\rbrace$, But I don't know how to prove it ?

Thanks for any help!

If $2k\in Z$, $k$ must be of the form $m/2$, where $m$ is an integer. Now investigate what would make $2/k$ an integer — David P, May 13 '21 at 16:44

score 2 · Accepted Answer · answered May 13 '21 at 16:49

2

Notice that $2k \cdot \frac{2}{k} = 4$, and since by assumption both $2k$ and $k/2$ are integers, you can see that $2k \in \{\pm 1, \pm 2, \pm 4 \}$ and hence $k$ is in the set you described.

answered May 13 '21 at 16:49

Sebastian Schulz

988

Thank you Sebastian for your answer ! – Mira May 13 '21 at 17:01

How to find the values of $k$ if $2k \in \mathbb{Z}$ and $\frac{2}{k}\in \mathbb{Z}$

1 Answers1