Question 92
The positive integers $a \gt b \gt 1$ are such that the equation
$$
\frac{a^x - 1}{a - 1} = \frac{b^y - 1}{b - 1}
$$
has at least two distinct solutions in positive integers $x \gt 1$ and $y \gt 1$. Prove that $a$ and $b$ are co-prime.
$$
\frac{a^x - 1}{a - 1} = \frac{b^y - 1}{b - 1}
$$
has at least two distinct solutions in positive integers $x \gt 1$ and $y \gt 1$. Prove that $a$ and $b$ are co-prime.