For every positive integer $b$, show that there exists a positive integer $n$ such that the polynomial ${x^2} - 1 \in (\mathbb{Z}/n\mathbb{Z})[x]$ has at least $b$ roots.
My efforts
Let $n = {p_1} \ldots {p_k}$, where ${p_i}$ is $i$-th prime number. Let us choose such $x$ for which $x-1$ is divisible by ${p_1} \ldots {p_i}$ and $x + 1$ is divisible by ${p_{i+1}} \ldots {p_k}$. The existence of such $x$ is guaranteed by CRT. Changing $i$ we get different solutions for $x$ modulo $n$. Indeed, if ${p_1} \ldots {p_j} \mid x-1$ and ${p_{j + 1}} \ldots {p_k} \mid x+1$ such that $i < j$, then both $x+1$ and $x-1$ are divisible by ${p_j} > 2$, which is impossible.