There are some nontrivial points that need to be addressed IMO. One, KM is not a thermodynamic dissociation constant; it can be greater than, equal to, or less than a dissociation constant in magnitude. Two, it is more fruitful to think in terms of what effect an inhibitor has on Vmax and Vmax/KM than it is helpful to ask what effect an inhibitor has on KM. (Leaving out subscripts) competitive inhibitors lower V/K but not V, whereas uncompetitive inhibitors lower V but not V/K. The apparent effects of the inhibitors on K are just by-products of these two effect. I may have more time this weekend to discuss further.