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The only external factors that impact K would be temperature because that can impact how much of the reactant is produced or product is created as well. Other factors such as the change in pressure or volume would not affect the K value because the system would adjust concentrations of products and reactants in the same specified ratio.
Furthermore, the change in concentration does not change K as the chemical reaction at equilibrium represents a fixed product to reactant ratio. The chemical reactions adjust so as to minimize the effect of changes. This is LeChatlier's Principle. K, however, is dependent on the temperature which is why we are always given a reaction at a specific temperature.
For a reaction in solution, both temperature and system pressure would affect the equilibrium constant. This is because both of these affect the chemical activity of a species. Since concentration is just an approximation of chemical activity, anything that changes chemical activity will change the equilibrium constant.
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