Say you have an integer that is produced by multiplying two random numbers
$$x_1 = a \cdot b_1 \bmod(p-1)$$ where $a$ and $b_1$ are relatively prime to $p-1$ and $p$ being a large prime.
Knowing $x_1$ leaves you with $p-1$ pairs of numbers for $a$ and $b_1$, so guessing the correct factorization is hard (right?).
Do you get any advantage in finding the correct pair if you get more $x_i$ where one of the factors is re-used? E.g.: $$x_2 = a \cdot b_2 \bmod(p-1)$$