In what number system is there an equality of the form $24=4 \cdot 6$ between three consecutive even or odd numbers?
In the $b$ number system, a number $a$ is to be found such that: $a b+(a+2)=(a+2)(a+4), b=\left(a^{2}+5 a+6\right) / a$. This implies that $a$ must be a divisor and thus $a$ is either 1, 2, 3, or 6. Accordingly, $b$ is either 12, 10, 10, or 12.

In the decimal system, such numbers are 24 and 35. In the duodecimal system: $1 \cdot 12^{1}+3 \cdot 12^{0}=3 \cdot 5$ and $6 \cdot 12^{1}+8 \cdot 12^{0}=$ $8 \cdot 10$.