  The present paper proposes a new parallel algorithm for the modular division
$u/v\bmod \beta^s$, where $u,\; v,\; \beta$ and $s$ are positive integers
$(\beta\ge 2)$. The algorithm combines the classical add-and-shift
multiplication scheme with a new propagation carry technique. This "Pen and
Paper Inverse" ({\em PPI}) algorithm, is better suited for systolic
parallelization in a "least-significant digit first" pipelined manner. Although
it is equivalent to Jebelean's modular division algorithm~\cite{jeb2} in terms
of performance (time complexity, work, efficiency), the linear parallelization
of the {\em PPI} algorithm improves on the latter when the input size is large.
The parallelized versions of the {\em PPI} algorithm leads to various
applications, such as the exact division and the digit modulus operation (dmod)
of two long integers. It is also applied to the determination of the periods of
rational numbers as well as their $p$-adic expansion in any radix $\beta \ge
2$.
