2008/09 AHSMC Part 1 Problem 11

Problem 11 of 16

In base $10$ , the positive integer $a$ has $2$ digits, the positive integer $b$ has $a$ digits and the positive integer $c$ has $b$ digits. The smallest possible value for $c$ is

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Solution

The smallest value of $a$ leads to the smallest value of $b$ , and of $c$ . The smallest value of $a$ is $10$ , a $1$ followed by $2-1$ $0$ s. The smallest value of $b$ is $10^9$ , a $1$ followed by $10-1$ $0$ s. The smallest value of $c$ is $10^{10^9-1}$ , a $1$ followed by $10^9-1$ $0$ s. The answer is (B).

University of Alberta

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