  A new interpretation and applications of the ``Diophantine'' and
factorisation properties of {\em finite} orthogonal polynomials in the Askey
scheme are explored. The corresponding twelve polynomials are the ($q$-)Racah,
(dual, $q$-)Hahn, Krawtchouk and five types of $q$-Krawtchouk. These
($q$-)hypergeometric polynomials, defined only for the degrees of
$0,1,\ldots,N$, constitute the main part of the eigenvectors of
$N+1$-dimensional tri-diagonal real symmetric matrices, which correspond to the
difference equations governing the polynomials. The {\em monic} versions of
these polynomials all exhibit the ``Diophantine'' and factorisation properties
at higher degrees than $N$. This simply means that these higher degree
polynomials are zero-norm ``eigenvectors'' of the $N+1$-dimensional
tri-diagonal real symmetric matrices. A new type of multi-indexed orthogonal
polynomials belonging to these twelve polynomials could be introduced by using
the higher degree polynomials as the seed solutions of the multiple Darboux
transformations for the corresponding matrix eigenvalue problems. The
shape-invariance properties of the simplest type of the multi-indexed
polynomials are demonstrated. The explicit transformation formulas are
presented.
