Suppose that m n a are in Z with gcdm n 1 and a 1 mod m a

Suppose that m, n, a are in Z with g.c.d(m, n) = 1 and a = 1 (mod m), a = 0 (mod n). Show that e = [a] is an idempotent of Z_mn different from [0] and [1].

Solution

Given e = [a] where e is identity of the group.

Let x = a mod mn

=> a = x mod mn

=> e = x mod mn

=> a² mod mn = e² mod mn = e mod mn (Because e is an identity and e² = e)

=> x = a mod mn = a² mod mn

=> [a]² = [a] = e

So [a] = e is an idempotent of Z_mn