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**Soroban** Hello, the undertaker!

Let the base be $\displaystyle b.$

Then $\displaystyle 25$ represents: .$\displaystyle 2b+5$

and $\displaystyle 66$ represents: .$\displaystyle 6b+6.$

Then the equation is: .$\displaystyle 3x^2 - (2b+5)x + (6b+b) \:=\:0$

Since $\displaystyle x = 4$ is a root: .$\displaystyle 3\cdot4^2 - (2b+5)\cdot4 + (6n+6) \:=\:0$

. . and we have: .$\displaystyle 48 - 8b - 20 + 6b + 6 \:=\:0 \quad\Rightarrow\quad -2b \:=\:-34 \quad\Rightarrow\quad b \;=\:17$

The problem was written in base-seventeen.