Determining if $c = 0 \mod N$ using the CRT?

Question

I was wondering if I'm misunderstanding the concept of the CRT theorem. So if we are given an expression to compute $c \mod N$ and check whether $c = 0 \mod N$ where $N = p.q$ two primes and $\log_2(N) = n$.

Can we compute the value of $c \mod u_1...u_k$ where $\log_2(u_1...u_k) > n$, and the $u_i$ are primes, using the CRT method. So if $c = 0 \mod u_1...u_k$, then $c = 0 \mod N$ since the size of $N$ is smaller than that of the primes product?

Answer 1

You can't just pick random primes $u_i$ that are large enough.

In order to check whether $c\equiv 0 \pmod{pq}$ you need to check that $c\equiv 0\pmod p$ and $c\equiv 0\pmod q$ specifically.