I am going to put it out here. VCAA has it wrong by having equilibrium constants with units. They DON'T have units. Otherwise the Gibbs free energy equation delta G = -RT ln K would make no sense. You can't take the natural log of something with units.
Strictly speaking, equilibrium constants relate chemical activities instead of concentrations, but we can usually make the approximation that the numerical value of the concentration in M or the partial pressure (yes, gas pressure not gas concentration) of the gas in bar (100 kPa) is equal to the chemical activity, which is dimensionless.
It's a pity VCE uses units -.-
In VCE language however, if you double the coefficients of reactants and products, the constant DOES change units. I'll give an example.
H+ + OH- <=> H2O
Let's assume that we have a neutral solution of [H+] = [OH-] = 10^-7 M
For this reaction (I don't need to state the states do I?), K = ([H+][OH-])^-1 = (10^-7 M * 10^-7 M)^-1 = 10^14 M^-2 at 298 K
Now look at 2H+ + 2OH- <=> 2H2O
Now K = ([H+]^2*[OH-]^2)^-1 = [10^-7 M]^-2 * [10^-7 M]*-2 = 10^28 M^4
And what do you know...the units have changed. It's because there's an exponential relationship between concentrations and reaction order, and if the sum of the indices of the top and bottom are different, then the units can change.