1993 AMC 12 第 7 题

Question

符号

R_k

表示一个十进制整数，它由连续

k

个数字一组成。例如，

R_3=111

、

R_5=11111

等。用

R_{24}

除以

R_4

，所得商

Q=\frac{R_{24}}{R_4}

是一个十进制表示中只含一和零的整数。

Q

中零的个数是

The symbol

R_k

stands for an integer whose base-ten representation is a sequence of

k

ones. For example,

R_3=111,

R_5=11111,

etc. When

R_{24}

is divided by

R_4,

the quotient

Q=\frac{R_{24}}{R_4}

is an integer whose base-ten representation is a sequence containing only ones and zeros. The number of zeros in

Q

is

10

11

12

13

15

使用

R_{24}=R_4(1+10^4+10^8+\cdots)

。

Use

R_{24}=R_4(1+10^4+10^8+\cdots)

商中有六个一，相邻两个一相隔四个位值。

The quotient has six ones, each four places apart

将这

24

个数字每四个分为一组，得到

\begin{aligned} R_{24} &=R_4\left(1+10^4+10^8\right.\\ &\qquad\left.{}+10^{12}+10^{16}\right.\\ &\qquad\left.{}+10^{20}\right) \end{aligned}\text{。}

因此

Q

中有六个一，每相邻两个一之间有三个零。所以共有

5\cdot3=15

个零。故正确答案是 E。

Grouping the

24

digits into six blocks of four gives

\begin{aligned} R_{24} &=R_4\left(1+10^4+10^8\right.\\ &\qquad\left.{}+10^{12}+10^{16}\right.\\ &\qquad\left.{}+10^{20}\right). \end{aligned}

Thus

Q

has six ones, with three zeros between each consecutive pair. It therefore has

5\cdot3=15

zeros. Thus the correct answer is E.

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