$K_c$ vs $K$ in equilibrium reactions
The thermodynamic equilibrium constant $K$ (different from both $K_p$ and $K_c$) and the reaction quotient $Q$ are defined with reference to a standard state. When all reactants and products are at standard state, $Q = 1$. $K$ and $Q$ are related to the Gibbs energy of reaction via:
$$\Delta_\mathrm{r}G = RT \ln\frac{Q}{K}$$ No matter which standard state you use, this relationship gives the same Gibbs energy of reaction (because $K$ and $Q$ are equally affected). However, depending on which standard state is used, $K$ and $Q$ will have different numeric values.

According to wikipedia,


  In chemistry, the standard state of a material (pure substance, mixture or solution) is a reference point used to calculate its properties under different conditions. In principle, the choice of standard state is arbitrary, although the International Union of Pure and Applied Chemistry (IUPAC) recommends a conventional set of standard states for general use.


The recommended standard state for a gas is, again according to wikipedia,


  the hypothetical state it would have as a pure substance obeying the ideal gas equation at standard pressure ($\pu{e5 Pa}$, or 1 bar). No real gas has perfectly ideal behavior, but this definition of the standard state allows corrections for non-ideality to be made consistently for all the different gases. 


Because neither the temperature nor the pressure is given in the problem the OP asked about, you can't use this standard state. Instead, you can use the standard state for a solute (still wikipedia):


  For a substance in solution (solute), the standard state is the hypothetical state it would have at the standard state molality or amount concentration but exhibiting infinite-dilution behavior. The reason for this unusual definition is that the behavior of a solute at the limit of infinite dilution is described by equations which are very similar to the equations for ideal gases. Hence taking infinite-dilution behavior to be the standard state allows corrections for non-ideality to be made consistently for all the different solutes. Standard state molality is 1 mol kg−1, while standard state amount concentration is 1 mol dm−3.


What is the problem with this problem?


  Am I wrong in the way I am approaching this problem, or is the problem itself wrong?


The problem is incomplete. If they use a funky standard state, they should say it explicitly.