2006 AMC 12A Problems/Problem 14
- The following problem is from both the 2006 AMC 12A #14 and 2006 AMC 10A #22, so both problems redirect to this page.
Two farmers agree that pigs are worth dollars and that goats are worth dollars. 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 dollar 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?
The problem can be restated as an equation of the form , where is the number of pigs, is the number of goats, and is the positive debt. The problem asks us to find the lowest x possible. p and g must be integers, which makes the equation a Diophantine equation. The Euclidean algorithm tells us that there are integer solutions to the Diophantine equation , where is the greatest common divisor of and , and no solutions for any smaller . Therefore, the answer is the greatest common divisor of 300 and 210, which is 30,
Alternatively, note that is divisible by 30 no matter what and are, so our answer must be divisible by 30. In addition, three goats minus two pigs give us exactly. Since our theoretical best can be achieved, it must really be the best, and the answer is . debt that can be resolved.
Let us simplify this problem. Dividing by , we get a pig to be: , and a goat to be . It becomes evident that if you exchange pigs for goats, we get the smallest positive difference - . Since we originally divided by , we need to multiply again, thus getting the answer:
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