2015 AMC 12A 第 15 题

Question

要把分数

\dfrac{123456789}{2^{26}\cdot 5^4}

表示成小数，小数点右边最少需要多少位数字？

What is the minimum number of digits to the right of the decimal point needed to express the fraction

\dfrac{123456789}{2^{26}\cdot 5^4}

as a decimal?

4

22

26

30

104

分子与分母没有共同的因子

2

或

5

。

The numerator shares no factor of

2

or

5

with the denominator

同时乘分子和分母，把分母变成

10

的幂

Multiply the top and bottom to turn the denominator into a power of

10

分子和分母没有公因数。要把这个分数写成小数，就把它改写成分母为

10

的幂的形式；最小可用的是

10^{26}

：

\dfrac{123456789}{2^{26}\cdot 5^4} = \dfrac{123456789\cdot 5^{22}}{10^{26}}\text{。}

因为分子

123456789\cdot 5^{22}

不能被

10

整除，所以小数点后恰好有

26

位。

因此，正确答案是 C。

The numerator and denominator share no common factors. To write the fraction as a decimal, rewrite it with a power of

10

in the denominator; the smallest that works is

10^{26}:

\dfrac{123456789}{2^{26}\cdot 5^4} = \dfrac{123456789\cdot 5^{22}}{10^{26}}.

Since the numerator

123456789\cdot 5^{22}

is not divisible by

10,

the decimal has exactly

26

places after the point.

Thus, the correct answer is C.

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