The Mod N Equi-divisor Classification of Integers

An equivalent divisor relation based on the modulus of integer n(mod n) is defined in the integer set Z. A mod n equi-divisor classification Z(n) of the integer n is therefore obtained. It is proved that the number of elements in Z(n) is T(n), where T(n) is the number of positive divisors of n.Based on the multiplication [a][b]=[ab], Z(n) forms a commutative semi-group, in which [0] is the zero element, [1] is the unit element and the only element having its inverse element. For the inequality T(n)+φ(n)≤ n+1, the equality holds true if and only if n=1,4p(p is a prime number and φ(n) is an Euler function).