Question 56
Let $a$ be a positive integer. For all positive integers $n$, we define
$$
a_n = 1 + a + a^2 + \dots + a^{n-1}.
$$
Let also $s, t$ be two different positive integers satisfying the following property: If $p$ is a prime divisor of $s-t$ then $p$ also divides $a-1$. Prove that the number
$$
\frac{a_s - a_t}{s - t}
$$
is an integer.
$$
a_n = 1 + a + a^2 + \dots + a^{n-1}.
$$
Let also $s, t$ be two different positive integers satisfying the following property: If $p$ is a prime divisor of $s-t$ then $p$ also divides $a-1$. Prove that the number
$$
\frac{a_s - a_t}{s - t}
$$
is an integer.