Maximum Common Modular Factor

The theory of modular factorizations was defined recently (in 2006). Since then, it has been characterized and studied by S. Hamon, R.M. Ortiz. The main goal of this work is to show the difference between the Greatest Common Divisor (GCD) and the Maximum Common Divisor (MCD) on modular, or τ_n-factorizations. Ortiz showed that the GCD on modular factorizations does not always exists. Ortiz and Luna are now defining the MCD of any two nonzero non-unit integers in the theory of τ_n-factorizations. Here, it will be discussed the MCD for the cases n=0 through n=4. This is a work in progress and we conjecture that the MCD always exist in the modular factorization theory.