2013 AMC 12A 第 17 题

Question

一组

12

名海盗同意按如下方式分一箱金币。第

k

个取份额的海盗拿走箱中剩余金币的

\dfrac{k}{12}

。箱中最初的金币数是能使每名海盗都得到正整数枚金币的最小数。第

12

名海盗得到多少枚金币？

A group of

12

pirates agree to divide a treasure chest of gold coins among themselves as follows. The

k

th pirate to take a share takes

\dfrac{k}{12}

of the coins that remain in the chest. The number of coins initially in the chest is the smallest number for which this arrangement will allow each pirate to receive a positive whole number of coins. How many coins does the

12

th pirate receive?

720

1296

1728

1925

3850

答案：D

知识点：整除性质因数分解阶乘

难度评级：2050

小提示：

在第

k

名海盗取走份额之前，箱中金币数是取走之后剩余金币数的

\dfrac{12}{12 - k}

倍

Before the

k

th pirate takes a share, the chest holds

\dfrac{12}{12 - k}

times what remains after

大提示：

若最后一名海盗拿到

n

枚金币，则初始金币数为

\dfrac{12^{11}n}{11!}

；令它成为整数并使

n

最小

If

n

coins remain for the last pirate, the initial count is

\dfrac{12^{11}n}{11!};

make it an integer with smallest

n

解答：

对

1 \le k \le 11

，第

k

名海盗取走份额之前的金币数，是取走之后金币数的

\dfrac{12}{12 - k}

倍。所以若留给第

12

名海盗的是

n

枚金币，初始金币数为

\dfrac{12^{11}\, n}{11!} = \dfrac{2^{14}\cdot 3^{7}\, n}{5^2\cdot 7\cdot 11}\text{。}

使其成为正整数的最小

n

为

5^2\cdot7\cdot11=1925

。在第

k

名海盗取金币之前，剩余的金币数等于初始数目乘以

\frac{11!}{(12-k)!\,12^{k-1}}

；代入这个

n

便可看出它对每个

k

都是整数。因此每个人分到的都是整数枚金币，而第

12

名海盗得到

1925

枚金币。

因此，正确答案是 D。

For

1 \le k \le 11,

the number of coins before the

k

th pirate takes a share is

\dfrac{12}{12 - k}

times the number afterward. So if

n

coins are left for the

12

th pirate, the initial count is

\dfrac{12^{11}\, n}{11!} = \dfrac{2^{14}\cdot 3^{7}\, n}{5^2\cdot 7\cdot 11}.

The smallest

n

making this a positive integer is

5^2\cdot7\cdot11=1925.

Before pirate

k,

the remaining count is the initial count multiplied by

\frac{11!}{(12-k)!\,12^{k-1}};

substituting this

n

shows it is an integer for every

k.

Hence all shares are integral, and the

12

th pirate receives

1925

coins.

Thus, the correct answer is D.

2013 AMC 12A 第 17 题

答案：D

难度评级：2050

小提示：

大提示：

解答：

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