Let $k_1<k_2<...<k_n$ be integer numbers. Prove that $$\det\big(V_n(1,2,..,n)\big)\mid \det\big(V_n(k_1,k_2,...,k_n)\big),$$ where V is a Vandermonde matrix.
I think it is a generalization of an easy problem: $$12\mid(a-b)(a-c)(a-d)(b-c)(b-d)(d-c),$$ and it's harder. I have no idea :(
Denote $(x)_k = x(x-1)\cdots (x-k+1)$ and consider the determinant $$ D = \begin{vmatrix} 1 & 1&\dots &1\\ {k_1\choose 1} & {k_2\choose 1}&\dots &{k_n\choose 1}\\ {k_1\choose 2} & {k_2\choose 2}&\dots &{k_n\choose 2}\\ \vdots & \vdots & \ddots & \vdots \\ {k_1\choose n-1} & {k_2\choose n-1}&\dots &{k_n\choose n-1} \end{vmatrix}. $$ Clearly, $$ D = \frac{1}{\det V_n(1,2,\dots,n)}\begin{vmatrix} 1 & 1&\dots &1\\ (k_1)_1 & (k_2)_1&\dots &(k_n)_1\\ (k_1)_2 & (k_2)_2&\dots &(k_n)_2\\ \vdots & \vdots & \ddots & \vdots \\ (k_1)_{n-1} & (k_2)_{n-1}&\dots &(k_n)_{n-1} \end{vmatrix}. $$ But the last determinant is easily transformed to $\det V_n(k_1,\dots,k_n)$ with the help of row operations.
It would be nice to see a combinatorial interpretation of $D$.