More Knowledge Integer Factorization 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.:
$$x_2 = a \cdot b_2 \bmod(p-1)$$
The question performs arithmetic in ${\mathbb Z_{p-1}}^*$, the integers reduced modulo $p-1$ that are coprime with this modulus. There are $\varphi(p-1)$ of these, where $\varphi$ is the Euler totient function. It is irrelevant that $p$ is prime, and the question has little to do with integer factorization.

For any $a\in{\mathbb Z_{p-1}}^*$, the application $\scr F_a:{\mathbb Z_{p-1}}^*\to{\mathbb Z_{p-1}}^*$, $b\to x=a\cdot b\bmod(p-1)$ is a permutation of ${\mathbb Z_{p-1}}^*$ (since $a$ has a multiplicative inverse). If follows that if $b$ is uniformly random on the set ${\mathbb Z_{p-1}}^*$, then $x$ is.

Therefore, no information is learned about $\scr F_a$, thus about $a$, from any number of $x_j$ disclosed without the corresponding $b_j$ (assumed independent and uniformly random on ${\mathbb Z_{p-1}}^*$), regardless of the size of parameters or computing power of adversary.