2003 AMC 10A Problem 20

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?

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Solution

The largest three-digit base-

9

number is

9^3 - 1 = 728,

and the smallest three-digit base-

11

number is

11^2 = 121.

So both conditions hold exactly when

121 \le n \le 728,

giving

608

integers. Out of

900

three-digit numbers, the probability is

\dfrac{608}{900} \approx 0.7.

Thus, the correct answer is E.

Tagged: number base · basic probability · counting integers in a range