Chinese Remainder Theorem for moduli that are not necessarily pairwise coprime

Assume that ${n_1},cdots ,{n_k}$ is a finite set of $k$ positive integers. We are not assuming that these integers are pairwise coprime. Consider the homomorphism of additive groups ${phi}:{bf Z}rightarrow {{bf Z}_{n_1}}times cdots times {{bf Z}_{n_k}}$ defined by $${phi}(x) = (xmod {n_1},cdots ,xmod{n_k}).$$
Is this mapping always surjective? If the moduli are pairwise coprime then we can use the Chinese Remainder Theorem to show that it must be surjective. But is this true in general?