Question

两位农民约定猪价值 $\$300$ ，山羊价值 $\$210$ 。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 $\$390$ 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 $\$300$ 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?

$\$5$

$\$10$

$\$30$

$\$90$

$\$210$

答案：C

知识点：最大公约数丢番图方程

难度评级：1580

小提示：

可结清的债务形如 $300p + 210g$ ，其中 $p, g$ 为整数。

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

大提示：

每个这样的值都是 $\gcd(300, 210)$ 的倍数。

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

解答：

债务 $D$ 能结清当且仅当 $D = 300p + 210g$ $= 30(10p + 7g)$ ，其中 $p, g$ 为整数。因此 $D$ 是 $\gcd(300, 210) = 30$ 的倍数，所以没有更小的正债务可行。

$\$30$ 的债务可以做到，因为 $30 = 300(-2) + 210(3)$ ，即给出 $3$ 只山羊并收到 $2$ 头猪作为找零。

因此，正确答案是 C。

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 第 14 题

答案：C

难度评级：1580

小提示：

大提示：

解答：

其他年份的第 14 题