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Given $ U, V, C $ are three distinct digits, figure them out from the following relation:

$$ \overline{CCC} = \overline{UVUVUV} \div \left ( \overline{UV} \times \overline{UV} \right) $$

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The solution is

$U = 1, V = 3, C = 7. $

Explanation:

Immediately, we can simplify the right side to $ \dfrac{10101}{\overline{UV}} $ because $ \overline{UV} | \overline{UVUVUV}. $ Since $ \overline{CCC} $ is divisible of 111, we should expect 10101 to be divisible by 111 as well. It is; we have $ 10101 = 111 \times 91, $ which gives us a possible solution of $ U = 9, V = 1, C = 1. $ However, this solution has two of the same digits, so we have to reject it. Luckily, we can factor 91 as $ 7 \times 13, $ giving us $ 10101 = 777 \times 13, $ which gives us our final solution.

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  • $\begingroup$ You say "immediately we can simplify...because xxx|xxxxxxx"; I don't know what that means, particularly the | symbol. Can you please explain? $\endgroup$ – Ken Y-N May 15 at 0:01
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    $\begingroup$ @KenY-N $ | $ means "divides into" or "is a factor of." The number $ \overline{UVUVUV} $ can be written as $ \overline{UV0000} + \overline{UV00} + \overline{UV}. $ Since all three terms are clearly divisible by $ \overline{UV} $ (they each factor into $ \overline{UV} $ times a power of 10), their sum is also divisible. $\endgroup$ – PiIsNot3 May 15 at 0:05

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