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The equilibrium constant is actually the activity of the product divided by the activity of the reactant. Calculating the activity is its own separate equation, but in that equation the units actually cancel. Therefore, the activity of the products and reactants have no units making K have no units too.
When you are looking for K is is molarity over molarity, which can cancel out all units. And if it doesn't cancel out all units, imagine the reactants changing to products or products changing to reactancts as activity, and activity is unitless.
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