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jmc

number theory senior

Problem

A base-10 three-digit number is selected at random. What is the probability that the base-9 representation and the base-11 representation of are both three-digit numerals?
Solution
The largest base-9 three-digit number is and the smallest base-11 three-digit number is . There are integers that satisfy , and 900 three-digit numbers altogether, so the probability is .
Final answer
\frac{152}{225}