Don't you have to show that:

s = n_k+1 (mod gcd(lcm(m1, m2, ..., mk), m_k+1))

is true before you can find n:

n = s (mod lcm(m_1, m_2, ..., m_k))
n = n_k+1 (mod m_k+1)

I can't figure out how to prove that the first eqn. is true.