ISL 2007 N4

For every integer $k \geq 2$, prove that $2^{3 k}$ divides the number $$\left(\begin{array}{c} 2^{k+1} \\ 2^{k} \end{array}\right)-\left(\begin{array}{c} 2^{k} \\ 2^{k-1} \end{array}\right)$$ but $2^{3 k+1}$ does not. (Poland)