2006 AMC 12A 第 14 题
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所有题目均经美国数学协会（MAA）官方合法授权使用。
14.
两位农民约定猪价值 ，山羊价值 。当一位农民欠另一位钱时，他用猪或山羊偿债，并可根据需要以山羊或猪的形式找零。（例如，一笔 的债可以用两头猪支付，并收到一只山羊作找零。）能用这种方式结清的最小正债务金额是多少？
Two farmers agree that pigs are worth and that goats are worth 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 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?
小提示：
可结清的债务形如 ，其中 为整数。
A resolvable debt has the form for integers
大提示：
每个这样的值都是 的倍数。
Every such value is a multiple of
解答：
债务 能结清当且仅当 ，其中 为整数。因此 是 的倍数，所以没有更小的正债务可行。
的债务可以做到，因为 ，即给出 只山羊并收到 头猪作为找零。
因此，正确答案是 C。
A debt is resolvable if and only if for integers Thus is a multiple of so no smaller positive debt works.
A debt of is achievable since i.e. give goats and receive pigs in change.
Thus, the correct answer is C.
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