Problem:
The town of Hamlet has people for each horse, sheep for each cow, and ducks for each person. Which of the following could not possibly be the total number of people, horses, sheep, cows, and ducks in Hamlet?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Let be the number of horses and be the number of cows. There are then people, ducks, and sheep in Hamlet. The total population of Hamlet is , where and are whole numbers. A number can be the population only if there exists a whole number value for such that is a whole number multiple of . This is possible for all the provided numbers except , as follows: , and
None of , and is a multiple of . Therefore cannot be the population of Hamlet.
Note: In fact, is the largest number that cannot be the population.
The problems on this page are the property of the MAA's American Mathematics Competitions