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Number theory Difficulty 4.3 AIME Find the answer Italy

Problem:

A merchant has 6 barrels with capacities 15, 16, 18, 19, 20 and 31 liters. Five of them are full of wine and only one of them is full of beer. The merchant keeps the barrel of beer for himself and sells all the barrels of wine to two different people, without splitting up their contents. If one of the two buyers bought a quantity of wine exactly double that bought by the other, how many liters does the barrel of beer contain?

Pick one

Solution

Solution:

The answer is (D). The sum of the capacities in liters of all the barrels is $119$ , a number that when divided by $3$ gives remainder $2$ . Since the sum of the liters of wine sold must be a multiple of $3$ , we must remove from the list of capacities the only value that when divided by $3$ gives remainder $2$ , that is $20$ . One then checks that the remaining barrels can indeed be divided as $15+18=33$ and $16+19+31=66$ .

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Source: MathNet, licensed CC-BY-4.0. Statement translated into English from it; metadata (topic, difficulty) added by this project.