Prove that if a set of even numbers do not divide into one another then doubling each of them preserves this property

Question

Assume I have a list of even integers where none can evenly divide into any of others. Is it true that if I multiply them all by $2$ then they still cannot divide into each other?

For example say my list is $12, 14, 22, 54$

None can dived another evenly.

If I multiply by $2$ I get $24, 28, 44, 108$

And still it is true.

Answer 1

If $a$ does not divide $b$, then $2a$ does not divide $2b$. This is simply because

$$\frac ba=\frac{2b}{2a}$$

If $\dfrac ba$ is not an integer, neither is $\dfrac{2b}{2a}$.

Answer 2

Suppose that $2n$ is divisible by $2m$. That means that $2n/2m = b$ where $b$ is an integer. Simplifying, that means that $n/m = b$ where $b$ is an integer, and that means that $n$ was already divisible by $m$ to begin with.

Answer 3

let $\{x_1,x_2,...x_n\}$ be a set of even numbers such that $x_i\not|x_j$ for all $i\neq j$. We want to show that $2x_i\not|2x_j$ for all $i\neq j$. Assume to the contrary that $i\neq j$ and $2x_i|2x_j$ then $\frac{2x_j}{2x_i}\in\Bbb{Z}$, but $\frac{2x_j}{2x_i}=\frac{x_j}{x_i}$, so $x_i|x_j$. This is a contradiction, so $2x_i\not|2x_j$ for all $i\neq j$.