For the reaction $\mathrm{N_2O_4(g)} \rightleftharpoons 2\,\mathrm{NO_2(g)}$ the relation between $K_p$ and $K_c$ is $K_p = K_c (RT)^{\Delta n}$ where $\Delta n = (\text{moles of gaseous products}) - (\text{moles of gaseous reactants}) = 2 - 1 = 1.$

Hence, $K_p = K_c (RT)$ so $K_c = \frac{K_p}{RT}.$

Substitute the given values: $K_p = 0.492, \quad R = 0.082\,\mathrm{L\,atm\,mol^{-1}\,K^{-1}}, \quad T=300\,\mathrm{K}$

$K_c = \frac{0.492}{0.082 \times 300}$

Calculate denominator: $0.082 \times 300 = 24.6$

Therefore, $K_c = \frac{0.492}{24.6} = 0.02 = 2 \times 10^{-2}.$

So the required integer in the blank is $\boxed{2}.$