Chinese remainder theorem. Applying it, however, involves Euclid's algorithm and requires some work (calculating an inverse modulo can be simplified a little using an iterative procedure described in this post). Wikipedia says that a solution exists sometimes when the divisors are not pairwise coprime, in particular, in this case, where all remainders are equal. Then a solution can be found using the method of successive substitution.
This problem is much easier because it has a single remainder 3 for all divisors. Therefore, 3 is a solution, though it is not in the given interval [200, 500]. The link for the Chinese remainder theorem above says that all solutions are congruent modulo the least common multiple of the divisors 8, 10, 12 and 30, i.e., 120. Therefore, the least solution in the given interval is 3 + 2 * 120 = 243.