Finding the coefficient of a variable given the coefficient of another

Question

I'm working with the following Kepler equation:

$$T=2\pi\sqrt{\frac{r^3}{GM}}$$

Which I have rearranged to find M:

$$M=\frac{r^3}{G(\frac{T}{2\pi})^2}$$

I have chosen the following values:

$r=100$
$M=100$
$T=76911340.145$
$G = 6.67408 × 10^{11}$

And now if I was to divide $r$ and $G$ by $1000$, I would like to know what coefficient must be applied to M to maintain the same value for T.

I have already managed to figure this out (somewhat) if I divide by $10$ instead: (I am looking for a general solution)

Given $r_2=\frac{r_1}{10}$, $G_2=\frac{G_1}{10}$, $M_2=\alpha M_1$:

$$\alpha=(\frac{1}{10})^2 \therefore M_2=(\frac{1}{10})^2M_1$$

And in more general terms:

$d=\frac{1}{10}, r_2=dr_1, G_2=dG_1, M_2=d^2M_1$

However my "proof" breaks down now when I use $d=\frac{1}{1000}$ and I do not know how to continue.

My attempt so far:

Long way:

$$M_2=\frac{(\frac{r_1}{1000})^3}{\frac{G_1}{1000}(\frac{T}{2\pi})^2}=1 × 10^{-5}$$

"Proof":

$$M_2=(\frac{1}{1000})^2M_1=1 × 10^{-4}$$

Clearly these two are not equal. Can somebody kindly please point me in the right direction?

Edit: Fixed question based on the results of the accepted answer.

Answer 1

I don't really see how you ended up with $d^3$ because it should be $d^2$:

$$2\pi\sqrt{\frac{(dr)^3}{(dG)(d^2M)}} = 2\pi\sqrt{\frac{d^3r^3}{d^3GM}} = 2\pi\sqrt{\frac{r^3}{GM}} = T.$$

Answer 2

If you divide $r$ and $G$ by $q=1000=10^3$ then $r^3/G$ will be divided by $(10^3)^3/10^3=10^6=q^2$. So you will get the same $T$ when you will divide $M$ by same number, that is $q^2=10^6$ (or multiplied by $10^{-6}$).