# What are letters doing with my numbers?

Number Theory Level 4

Consider a three digit number where:

$${ A }^{ 3 }+{ B }^{ 3 }+{ C }^{ 3 }+ABC\quad =\quad \overline { ABC }$$

$$\frac { A+2B+4C }{ 3 } =A$$

What is $$\overline {ABC}$$ ?

Details and Assumptions: $$\overline {ABC}$$ is a three digit number, whereas ABC indicates multiplication. $$\overline {ABC}$$ is positive.
NOTE: A, B, and C are the digits of the number $$\overline {ABC}$$. By that definition, A, B, and C are integers from 0 to 9.

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