AMC10 2006 A

AMC10 2006 A · Q22

AMC10 2006 A · Q22. It mainly tests GCD & LCM, Diophantine equations (integer solutions).

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?

两位农民约定猪价值 $300$，羊价值 $210$。当一位农民欠另一位钱时，他用猪或羊支付债务，并必要时以羊或猪形式收到找零。（例如，$390$ 的债务可以用两头猪支付，并收到一头羊作找零。）可以用这种方式解决的最小正债务金额是多少？

(A) $5 $5

(B) $10 $10

(D) $90 $90

(E) $210 $210

Answer

Correct choice: (C)

正确答案：（C）

Solution

(C) If a debt of $D$ dollars can be resolved in this way, then integers $p$ and $g$ must exist with $D = 300p + 210g = 30(10p + 7g)$. As a consequence, $D$ must be a multiple of 30, and no positive debt less than \$30 can be resolved. A debt of \$30 can be resolved since $30 = 300(-2) + 210(3)$. This is done by giving 3 goats and receiving 2 pigs.

（C）如果一笔金额为 $D$ 美元的债务可以用这种方式解决，那么必定存在整数 $p$ 和 $g$，使得 $D = 300p + 210g = 30(10p + 7g)$。因此，$D$ 必须是 30 的倍数，并且任何小于 \$30 的正债务都无法解决。\$30 的债务可以解决，因为 $30 = 300(-2) + 210(3)$。这可以通过给出 3 只山羊并收到 2 只猪来完成。

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