Question 27
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.
$$
\left(\begin{array}{c}
2^{k+1} \\
2^{k}
\end{array}\right)-\left(\begin{array}{c}
2^{k} \\
2^{k-1}
\end{array}\right)
$$
but $2^{3 k+1}$ does not.