What is the next smallest number of coins that would satisfy the conditions of Problem 59?

Data from Problem 59

Chinese Remainder Problem A band of 17 pirates decided to divide their doubloons into equal portions. When they found that they had 3 coins remaining, they agreed to give them to their Chinese cook, Wun Tu. But 6 of the pirates were killed in a fight. Now when the treasure was divided equally among them, there were 4 coins left that they considered giving to Wun Tu. Before they could divide the coins, there was a shipwreck and only 6 pirates, the coins, and the cook were saved. This time equal division left a remainder of 5 coins for the cook. Now Wun Tu took advantage of his culinary position to concoct a poison mushroom stew so that the entire fortune in doubloons became his own. What is the smallest number of coins that the cook would have finally received?