  In his celebrated book "Mathematical Treatise in Nine Sections" of 1247, Qin,
Jiushao described the Chinese remainder theorem with great detail and
generality. He also gave a method for computing modular inverse under the name
of "DaYan deriving one". Historical significance of DaYan deriving one method
has been well studied. In this paper, we investigate its modern mathematical
nature from the perspectives of number theory and algorithm. One of the
remarkable features of Qin's algorithm is that it keeps a state of four
variables in a matrix form. Its choice of variables and layout provide natural
ways of connecting several important mathematical concepts. An invariant about
the state is also observed which provides a convenient yet powerful tool in
proving several important mathematical results. The paper first explains Qin's
algorithm and proves some of its properties. Then the connection with continued
fractions is examined, the results show that the states of Qin's algorithm
contain rich information about continued fractions and some classical arguments
can be derived easily. The last part of the paper discusses a family of
2-dimensional lattices of number theoretic significance by proving that the
shortest vectors of these lattices can be obtained from the states of Qin's
algorithm. This result is surprising in that a shortest lattice vector is found
in a well-regulated set. A method of computing such shortest vectors is
proposed.ed.
