In my physics book, they consider a positive point charge $q$ and a spherical surface of radius $r$ centered on this positive charge. Then the flux through this surface is given by $$\Phi = \iint \vec{E} \cdot d\vec{A} = \iint EdA\cos\theta$$
Since $\vec{E}$ and $\vec{A}$ are parallel, $\cos\theta=1$, so the integral become just $$\iint EdA$$ Furthermore, $E$ is constant on every point so you get $$\iint EdA = E \iint dA = E(4\pi r^2)$$ where the last step follows from the fact that the surface is a sphere. Then according to Gauss's law $E=kq/r^2$ we get $$\Phi = \left(\frac{kq}{r^2}\right)(4\pi r^2) = 4\pi kq$$
Then finally they define a new constant $\epsilon_0$ which is equal to $1/(4\pi k)$ so that we get $$\iint\vec{E}\cdot d\vec{A} = \frac{q_{enclosed}}{\epsilon_0}$$
Now the real question: why is $\epsilon_0 = 1/(4\pi k)$

Why $\epsilon_0 = 1/(4\pi k)$ instead of $\epsilon_0 = 4\pi k$? Why use the inverse (reciprocal)? Whouldn't be $$\iint\vec{E}\cdot d\vec{A} = q_{enclosed} \epsilon_0$$ an "easier" and better choice?

Thanks in advance.

Edit: With easier I mean a more logical choice.

Edit 2: I was under the impression that $\epsilon_0$ is defined to make Gauss's law simpler. I voted this question to close.