Question

要把分数表示成小数，小数点右边最少需要多少位数字？ $\dfrac{123456789}{2^{26}\cdot 5^4}$

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?

$4$

$22$

$26$

$30$

$104$

答案：C

知识点：小数质因数分解

难度评级：1800

解答：

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

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.

2015 AMC 12A 第 15 题

答案：C

难度评级：1800

解答：

其他年份的第 15 题