AMC10 2020 B

AMC10 2020 B · Q12

AMC10 2020 B · Q12. It mainly tests Decimals, Base representation.

The decimal representation of $\frac{1}{20^{20}}$ consists of a string of zeros after the decimal point, followed by a 9 and then several more digits. How many zeros are in that initial string of zeros after the decimal point?

小数表示 $\frac{1}{20^{20}}$ 在小数点后有一串零，然后是一个 9，后面还有几个数字。小数点后那串初始零有多少个？

(A) 23 23

(B) 24 24

(D) 26 26

(E) 27 27

Answer

Correct choice: (D)

正确答案：（D）

Solution

Answer (D): Observe that $20^{20}=2^{20}\cdot 10^{20}=1024^{2}\cdot 10^{20}$. Because $10^{3}<1024<2000<10^{3.5}$, it follows that $10^{6}<1024^{2}<10^{7}$. Therefore $\frac{1}{20^{20}}$ is strictly between $10^{-27}$, whose decimal representation consists of 26 zeros after the decimal point followed by a 1, and $10^{-26}$, whose decimal representation consists of 25 zeros after the decimal point followed by a 1. Hence there are 26 zeros to the right of the decimal point preceding the first nonzero digit. The exact value is in fact \[ \frac{1}{20^{20}}=0.00000\ 00000\ 00000\ 00000\ 00000\ 09536\ 74316\ 40625. \]

答案（D）：注意到 $20^{20}=2^{20}\cdot 10^{20}=1024^{2}\cdot 10^{20}$。因为 $10^{3}<1024<2000<10^{3.5}$，所以 $10^{6}<1024^{2}<10^{7}$。因此 $\frac{1}{20^{20}}$ 严格介于 $10^{-27}$ 与 $10^{-26}$ 之间：$10^{-27}$ 的十进制表示是在小数点后有 26 个 0 然后是 1；而 $10^{-26}$ 的十进制表示是在小数点后有 25 个 0 然后是 1。于是，小数点右侧在第一个非零数字之前有 26 个 0。其精确值实际上是 \[ \frac{1}{20^{20}}=0.00000\ 00000\ 00000\ 00000\ 00000\ 09536\ 74316\ 40625. \]

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