Question about modular arithmetic (division)
The answer depends on how $n$ is chosen.

If $n$ is coprime with each of the $y_j$ (including if $n$ is prime and all the $y_j$ are in range $[1..n[$), and $x_1\bmod n$ is uniformly random on $[0..n[$ (including if $x_1$ is uniformly random on that interval, or on a much larger interval), then demonstrably, Bob can not determine what $y_i$ Alice has picked by examination of $X\;=\;x_1*y_i\bmod n$. Proof sketch: the application $F_i:x_1\to X$ is a permutation of $[0..n[$; therefore $X$ is uniformly random on $[0..n[$, regardless of $y_i$; thus $X$ can't leak information about $y_i$.

However, if $n$ is the product of all the $y_i$ and these are distinct moderate primes, then Bob can find with fair odds which $y_i$ was chosen: $X$ is divisible by that $y_i$, and only those few (perhaps no) other $y_j$ that divide $X$.

There are gradations between theses two extremes.