On divisibility of binomial coefficients (Q1343279)

Let \(p\) be a prime and \(A(n,p)\) the \(p^ n\times p^ n\)-matrix with entries \(a_{ij}= \left(\begin{smallmatrix} i\\ j\end{smallmatrix}\right)\text{ mod }p\) for \(0\leq i,j< p^ n\). It is shown that \(A(n,p)\) is the \(n\)-fold tensor product of \(A(1,p)\) with itself. As an application a short proof is given that there are precisely \(\left(\begin{smallmatrix} p+1\\ 2\end{smallmatrix}\right)^ n\) binomial coefficients \(\left(\begin{smallmatrix} i\\ j\end{smallmatrix}\right)\) for \(0\leq i< p^ n\) which are not divisible by \(p\).