Z,(2/2) (Z X)is not a group,because 0-1 does not exist. Even if we exclude 0 and consider only Z=Z0), (Z,X)is not necessarily a group;some a!may not exist. For a Z exists if and only if gcd(a,n)=1 gcd(a,n)=1>there exists integers x and y s.t. ax+ny=1 a][x][n][y][1]in Z →[a[x]=[l]inZm →[a]l=[x]inZm 1010 Zn (2/2)  (Zn, ×) is not a group, because 0-1 does not exist.  Even if we exclude 0 and consider only Zn + = Zn\{0}, (Zn + , ×) is not necessarily a group; some a-1 may not exist.  For a ∈Zn, a-1 exists if and only if gcd(a, n)=1  gcd(a, n) = 1 ⇔ there exists integers x and y s.t. ax + ny = 1 ⇔ [a][x] + [n][y] = [1] in Zn ⇔ [a][x] = [1] in Zn ⇔ [a]-1 = [x] in Zn