Perhaps I'm missing something, but G & c are both typically constants, so they can't be just set to be equal to 1.

Typically we count in base 10 (numbers 0,1,2,3,4,5,6,7,8,9,10,...)

Computers work in binary (base 2) and various conversion schemes (0,1,10,11,100,101,110,111,1000,1001...)

One can also count in octal (0,1,2,3,4,5,6,7,10,11,12,13,14,15,16,17,20,...)

Or hexidecimal (0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F,10,11,12,13,14,15,16,17,18,19,1A,1B,1C,...)

Technically the base can be any number one wants, although the conversion to an irrational number would be tricky, but not technically impossible.

Distance is often written in many different units. Don't like one, try another.

Angstrom

microns, nm, mm, cm, meters, km, etc.

AU

light year

and etc.

Changing your units, and you can simplify things, as long as you don't get too confused with the units.

For example, if you wished to know the area of the circle inscribed by Earth's orbit, πr^{2} (assuming a circular orbit). If you do the calculations in meters, one gets a really big number.

However, in AU, it is simply π(1AU)^{2} = πAU^{2}

I suppose since c is expressed in meters and seconds.

and G is expressed in kg, meters, and seconds.

Then one could transform it so one now expresses c in a new unit, say CM which is defined as 3x10^{8}m, so the speed of light is now 1 CM/s.

Then you define G in terms of your new CM unit, time, and some funky weight unit.

But, of course, you have to always maintain the units as part of your equation, and the calculations come out the same.