Question 19
Let $a \gt 2$ be an integer. Let
$$x = (a-1) \cdot a^{a-2} + (a-2) \cdot a^{a-3} + \dots + 2 \cdot a^1 + 1 \cdot a^0,$$
$$y = 1 \cdot a^{a-2} + 2 \cdot a^{a-3} + \dots + (a-2) \cdot a^1 + (a-1) \cdot a^0.$$
Prove that $x - 1$ is divisible by $y + 1$.
$$x = (a-1) \cdot a^{a-2} + (a-2) \cdot a^{a-3} + \dots + 2 \cdot a^1 + 1 \cdot a^0,$$
$$y = 1 \cdot a^{a-2} + 2 \cdot a^{a-3} + \dots + (a-2) \cdot a^1 + (a-1) \cdot a^0.$$
Prove that $x - 1$ is divisible by $y + 1$.