Prove that a given expression is always an integer 
Given integers $x_1, x_2, \dotsc, x_n$, prove that the expression
  $$
  \prod \limits_{1\leq i<j\leq n}\frac{x_i - x_j}{i-j}
$$
  is always an integer. 

I think induction should work, but I couldn't get anywhere with it.
 A: I don't know of a really easy of proving this fact, but here is one possible proof. The product $\prod_{1\leq i<j\leq n}(i-j)=\prod_{i=1}^ni!$ is a super factorial. The idea is that $\prod_{1\leq i<j\leq n}(x_i-x_j)$ is a vandermonde determinant, i.e. a determinate of the matrix:
$$\begin{pmatrix} 1 & x_1 & x_1^2\cdots & x_1^{n-1}\\ 1 & x_2 & x_2^2\cdots & x_2^{n-1}\\
\vdots & \vdots & \vdots & \vdots\\
1 & x_n & x_n^2\cdots & x_n^{n-1}
\end{pmatrix}.$$
The key is that by applying column operations to the above matrix, we can transform it the determinant of:
$$\begin{pmatrix} 1 & p_1(x_1) & p_2(x_1)\cdots & p_{n-1}(x_1)\\ 1& p_1(x_2) & p_2(x_2)\cdots & p_{n-1}(x_2)\\
\vdots & \vdots & \vdots & \vdots\\
1 & p_1(x_n) & p_2(x_n)\cdots & p_{n-1}(x_n)
\end{pmatrix},$$
where $p_i(x)$ is any collection of polynomials that are monic (having leading coefficient 1). In this case you can choose $p_i(x)=x(x-1)\cdots (x-i+1)=\binom{x}{i}i!$. Thus the product $\prod_{i=1}^ni!$ factors out and you're left with a determinant of binomial coefficients, which is an integer. 
