For every integer
![k \geq 2,](/media/m/0/3/d/03d6ac75d711058abed140719184e106.png)
prove that
![2^{3k}](/media/m/e/3/1/e316c1e867f5f32e2fa1b0cc40f784f5.png)
divides the number
but
![2^{3k + 1}](/media/m/f/5/0/f50844cb9748d6898d9bb2ec3e816645.png)
does not.
Author: unknown author, Poland
%V0
For every integer $k \geq 2,$ prove that $2^{3k}$ divides the number
$$\binom{2^{k + 1}}{2^{k}} - \binom{2^{k}}{2^{k - 1}}$$
but $2^{3k + 1}$ does not.
Author: unknown author, Poland