TITLE: Is $\prod_{1\leq i< j\leq n} \frac{a_i - a_j}{i-j}$, with distinct integers $a_i$, an integer?
QUESTION [16 upvotes]: It is known that for every $n$ consecutive integers, their product is divisible by $n!$, since $${{m}\choose{n}} = \frac{m!}{n!(m-n)!}$$ is also an integer.
So is it true that for every distinct integer $a_1, a_2, ..., a_n$, the number $$S = \prod_{1\leq i< j\leq n} \frac{a_i - a_j}{i-j}$$ is also an integer?
(Sorry for my grammar mistakes, English is my second language)

REPLY [15 votes]: We observe that $\prod_{1 \leq i < j \leq n} (i-j) = \prod_{r=1}^n r!$ and the numerator is just the Vandermonde Determinant so that by performing row operations we obtain
$$\prod_{1 \leq i < j \leq n} \frac{a_i-a_j}{i-j} = 
\begin{vmatrix}
1 & 1 & \cdots & 1 \\
\binom{a_1}{1} & \binom{a_2}{1} & \cdots & \binom{a_n}{1} \\
\binom{a_1}{2} & \binom{a_2}{2} & \cdots & \binom{a_n}{2} \\
\cdots & \cdots & \cdots & \cdots \\
\binom{a_1}{n-1} & \binom{a_2}{n-1} & \cdots & \binom{a_n}{n-1} \\
\end{vmatrix}$$
which is obviously an integer.