Question

Two farmers agree that pigs are worth $\$300$ and that goats are worth $\$210.$ When one farmer owes the other money, he pays the debt in pigs or goats, with “change” received in the form of goats or pigs as necessary. (For example, a $\$390$ debt could be paid with two pigs, with one goat received in change.) What is the amount of the smallest positive debt that can be resolved in this way?

$\$5$

$\$10$

$\$30$

$\$90$

$\$210$

Answer: C

Concepts:greatest common divisor Diophantine Equation

Difficulty rating: 1580

Small Hint:

A resolvable debt has the form $300p + 210g$ for integers $p, g$

Big Hint:

Every such value is a multiple of $\gcd(300, 210)$

Solution:

A debt $D$ is resolvable if and only if $D = 300p + 210g$ $= 30(10p + 7g)$ for integers $p, g.$ Thus $D$ is a multiple of $\gcd(300, 210) = 30,$ so no smaller positive debt works.

A debt of $\$30$ is achievable since $30 = 300(-2) + 210(3),$ i.e. give $3$ goats and receive $2$ pigs in change.

Thus, the correct answer is C.

2006 AMC 12A Problem 14

Answer: C

Difficulty rating: 1580

Small Hint:

Big Hint:

Solution:

Problem 14 in Other Years