Solution:
As per the question,
$Y=\left\{x \mid x\right.$ is a positive factor of the number $2^{p-1}\left(2^{p}-1\right)$, in which $2^{p}-1$ is a prime number $\}$.
Roster form,
1 and p itself are the only possible positive factors of a prime number p.
The possible factors of $2^{p-1}\left(2^{p}-1\right)$ are all possible factors of $2^{p-1}$ and $2^{p}-1$ individually.
The possible factors of $2^{p-1}$ are $2^{0}, 2^{1} \ldots 2^{p-1}$ and that of $2^{p}-1$ are 1 and $2^{p}-1\left\{\because ~{{2}^{p}}~\text{-}1\text{ }is\text{ }prime\text{ }number\}$
Hence,
$x=1,2^{1} \ldots 2^{p-1}, 2^{p}-1$
As a result, $Y=\left\{1,2^{1} \ldots 2^{p-1}, 2^{p}-1\right\}$