We know that if we have information on $$N\bmod p_i$$ where $p_i$ are primes at every $i\in\{1,\dots,m\}$ where $m\in\Bbb N$ is such that $\prod_{i=1}^mp_i<N$, then we cannot reconstruct $N$ by CRT. We need to have $\prod_{i=1}^mp_i>N$.
My simple query is if $m$ is such that $\prod_{i=1}^mp_i\gg N$ then do we have any ambiguity in reconstruction? Or do any $m$ such that $\prod_{i=1}^mp_i>N$ suffices?
In general how do we reconstruct $N$ assuming number of bits in $N$ is known?