Consider a quadratic equation $x^2 - 13x + 36 = 0$ with coefficients in a base $b$. The solutions of this equation in the same base $b$ are $x = 5$ and $x = 6$. Then $b =$ ________.

For the quadratic equation $x^2 - 13_b x + 36_b = 0$ with roots $x = 5_b$ and $x = 6_b$:

1. The sum of the roots is equal to the coefficient of the linear term:

   $$5 + 6 = 13_b$$

   In decimal, $5 + 6 = 11$, and $13_b = 1 \cdot b + 3 = b + 3$.

   $$b + 3 = 11 \implies b = 8$$

2. We can verify using the product of the roots:

   $$5 \times 6 = 36_b$$

   In decimal, $5 \times 6 = 30$, and $36_8 = 3(8) + 6 = 24 + 6 = 30$.

Both conditions are consistently satisfied for $b = 8$.