Representing extended as a modula equation

Question

I've recently being doing research on discrete arithmetic, brought about by a research in artificial intelligence algorithm that analyses data much more accurately as rounding of digits is eliminated and each information is handled as only quotients and remainders unless otherwise with some more complex advantages. But am stack at representing discrete algorithms as equations most specifically modulo inverse instead am getting an unknown number of system of equation, forcing me to represent as a sum at the cost of performance but with good intelligence increase and complex pattern recognition. i need help on this please. Note that finding modulo inverse $\mathbb a_ix+N_iy=\gcd(N_i,a_i)=1$ where $\mathbb (N_i)\mod N=0$ and $a_i \mod N=a$.