# It is as easy as A B C, Figure out U V C from the given relationship

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

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

$$U = 1, V = 3, C = 7.$$
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.
• You say "immediately we can simplify...because xxx|xxxxxxx"; I don't know what that means, particularly the | symbol. Can you please explain? May 15 '19 at 0:01
• @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.