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?

$41$

$47$

$59$

$61$

$66$

Answer: B

Concepts:Diophantine Equation modular arithmetic

Difficulty rating: 1420

Solution:

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.

2015 AMC 10B Problem 15

Answer: B

Difficulty rating: 1420

Solution:

Problem 15 in Other Years