The base (or radix) of the number system such that the following equation holds is __________. $$\frac{312}{20} = 13.1$$

Let the unknown base be $b$.

Convert each term in base $b$ to decimal representation:

• Numerator: $(312)_b = 3b^2 + 1b + 2$

• Denominator: $(20)_b = 2b + 0 = 2b$

• Right-hand side: $(13.1)_b = 1b + 3 + 1b^{-1} = b + 3 + \frac{1}{b} = \frac{b^2 + 3b + 1}{b}$

Set up the equation:

Multiply both sides by $2b$ (since $b > 0$):

Since the digits used include $3$, the base must satisfy $b > 3$. Hence, $b = 5$.