AMC10 2003 A

AMC10 2003 A · Q20

AMC10 2003 A · Q20. It mainly tests Probability (basic), Base representation.

A base-10 three-digit number $n$ is selected at random. Which of the following is closest to the probability that the base-9 representation and the base-11 representation of $n$ are both three-digit numerals?

随机选取一个十进制的三位数 $n$。$n$ 的九进制表示和十一进制表示均为三位数表示法的概率最接近于以下哪一项？

(A) 0.3 0.3

(B) 0.4 0.4

(D) 0.6 0.6

(E) 0.7 0.7

Answer

Correct choice: (E)

正确答案：（E）

Solution

(E) The largest base-9 three-digit number is $9^3-1=728$ and the smallest base-11 three-digit number is $11^2=121$. There are $608$ integers that satisfy $121\le n\le 728$, and $900$ three-digit numbers altogether, so the probability is $608/900\approx 0.7$.

（E）九进制的最大三位数是 $9^3-1=728$，十一进制的最小三位数是 $11^2=121$。满足 $121\le n\le 728$ 的整数有 $608$ 个，而所有三位数一共有 $900$ 个，所以概率是 $608/900\approx 0.7$。

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