Question

两位农夫约定猪值 $\$300$ ，山羊值 $\$210$ 。当一位农夫欠另一位钱时，他用猪或山羊偿还，并可按需要以山羊或猪的形式收取“找零”。（例如， $\$390$ 的债可以用两头猪支付，并收回一只山羊作为找零。）用这种方式可以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：1630

解答：

可结清的债务可写为 $D = 300p + 210g$ ，其中 $p, g$ 为整数，负值表示收到找零。因为 $D = 30(10p + 7g)$ ，且 $\gcd(10, 7) = 1$ ，所以 $10p + 7g$ 可以是任意整数，故 $D$ 可以是任意 $30$ 的倍数。

最小的正数是 $30$ ，因为 $30 = 300(-2) + 210(3)$ ，也就是交出 $3$ 只山羊并找回 $2$ 只猪。

所以正确答案是 C。

A resolvable debt is $D = 300p + 210g$ for integers $p, g,$ where a negative value means change received. Since $D = 30(10p + 7g)$ and $\gcd(10, 7) = 1,$ the value $10p + 7g$ can be any integer, so $D$ is any multiple of $30.$

The smallest positive one is $30,$ achieved by $30 = 300(-2) + 210(3)$ (give $3$ goats, receive $2$ pigs).

Thus, the correct answer is C.

2006 AMC 10A 第 22 题

答案：C

难度评级：1630

解答：

其他年份的第 22 题