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Number theory Difficulty 4.9 AIME Find the answer Italy

Problem:

A number is called "modern" if, in base 10, it can be expressed by concatenating "a certain number" of decimal representations of $2006$ : for example $200620062006$ is modern, while $20200606$ and $2006200$ are not. How many digits does the smallest positive modern perfect square have?

Pick one

Solution

Solution:

The answer is (E). A modern number is always divisible by $2006$ , and the quotient is of the form $1000100010001 \ldots 10001$ , hence odd. A modern number is therefore always even, but is never divisible by $4$ , and thus cannot be a perfect square.

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Source: MathNet, licensed CC-BY-4.0. Statement translated into English from it; metadata (topic, difficulty) added by this project.