7. Find all three-digit numbers in the decimal system that are equal to one third of the number with the same representation in another number system.
SOLUTION. The sought number in the decimal system has the form

$$
100 A+10 B+C, \quad A, B, C \in\{0,1, \ldots, 9\}, A \neq 0
$$

If $z$ is the unknown base of another number system, according to the condition of the problem, we have

$$
100 A+10 B+C=\frac{1}{3}\left(A z^{2}+B z+C\right),
$$

from which we get the equation

$$
A\left(z^{2}-300\right)=B(30-z)+2 C .
$$

Clearly, $z \geq 18$, because $17^{2}<300$. Let's consider individual cases:

If $z=18$, then $12 A=6 B+C$. The solutions are these triples $(A, B, C):(1,2,0)$, $(1,1,6),(2,4,0),(2,3,6),(3,6,0),(3,5,6),(4,8,0),(4,7,6),(5,9,6)$.

If $z=19$, then $61 A=11 B+2 C$. The solution is the triple $(A, B, C)=(1,5,3)$.

If $z=20$, then $50 A=5 B+C$. The solution is the triple $(A, B, C)=(1,9,5)$.

For $z \geq 21$, the equality will not occur for any triple $(A, B, C)$, because the left side of the equation is always greater than or equal to 141 and the right side is always less than or equal to 99.

In total, 11 numbers satisfy the condition: 116, 120, 153, 195, 236, 240, 356, 360, 476, 480, and 596.

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