Which barrel holds the beer?

Question

A merchant has six barrels holding 15, 16, 18, 19, 20 and 31 gallons. Five contain wine and one contains beer. He sells the wine, in whole barrels, to two customers: the first buys some barrels and the second buys exactly twice as many gallons as the first. All five wine barrels are sold. Which barrel contains the beer, and who bought what?

Accepted Answer

The 20-gallon barrel holds the beer. The first customer bought the 15 and 18 gallon barrels (33 gallons); the second bought the 16, 19 and 31 gallon barrels (66 gallons). Divisibility by three. The first customer bought W gallons and the second 2W, so the wine sold is 3W, a multiple of 3. All six barrels total 15 + 16 + 18 + 19 + 20 + 31 = 119. If the beer barrel holds B gallons, the wine is 119 minus B, which must be a multiple of 3. Since 119 leaves remainder 2 when divided by 3, B must also leave remainder 2. Check the barrels: 15 (remainder 0), 16 (1), 18 (0), 19 (1), 20 (2), 31 (1). Only…

Which barrel holds the beer?

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