Question 24

For a quadratic equation $x^2 - c_1 x + c_0 = 0$ with roots $r_1 = 5$ and $r_2 = 6$ in base $b$:

• Sum of roots $= r_1 + r_2 = 5 + 6 = 11_{10}$.
• Product of roots $= r_1 \times r_2 = 5 \times 6 = 30_{10}$.

From the equation $x^2 - 13_b x + 36_b = 0$:

1. The coefficient of $x$ is $13_b = 1 \cdot b + 3 = b + 3$.
   Equating to the sum of roots:
   $$b + 3 = 11 \implies b = 8$$
2. Check the constant term with $b = 8$:
   $$36_8 = 3 \times 8 + 6 = 24 + 6 = 30_{10}$$
   This matches the product of roots $5 \times 6 = 30_{10}$ perfectly.

Hence, the base is $b = 8$.