2015 AMC 10B Problem 15

Question

The town of Hamlet has

3

people for each horse,

4

sheep for each cow, and

3

ducks for each person. Which of the following could not possibly be the total number of people, horses, sheep, cows, and ducks in Hamlet?

41

47

59

61

66

Write the total as

13h+5c

Check each option modulo

5

after subtracting multiples of

13

If there are

h

horses and

c

cows, then there are

3h

people,

9h

ducks, and

4c

sheep. The total is therefore

13h+5c

.

The listed values except

47

can be written in that form:

\begin{gathered} 41=13\cdot2+5\cdot3, \\ 59=13\cdot3+5\cdot4, \\ 61=13\cdot2+5\cdot7, \\ 66=13\cdot2+5\cdot8. \end{gathered}

For

47

, subtracting

0,13,26,39

leaves

47,34,21,8

, none of which is divisible by

5

.

Thus, the correct answer is B.

Problem 15 in other years