The commander of an army of ancient Rome decides to give each of his soldiers a commemorative vessel on the anniversary of the conquest of Gaul. So instruct the local potter to make the vessels and have them finished on the anniversary day. If the potter makes 25 vessels every day, he would have 15 left at the end of the term, but if he does 26, he will have 8 left over at the end of the term.

**How many soldiers does the army have?**

#### Solution

Since the difference between the first case and the second one is only one more vessel per day, we can know how many vessels there are of more, because it would manufacture the 15 that are needed for the order and 8 more, that is, 23 vessels more than In the first case. Then there must be 23 days left before the celebration. If you manufacture 25 vessels a day, you will have 575 and since you are supposed to be 15, the total number of vessels will be **590 that matches the total soldiers**.