I'm stuck on the following exercise:
Let $N$ be a secret that we want to share among $k$ people such that any subset of $k'\leq k$ people can reconstruct $N$ but any $k'-1$ people cannot find the secret $N$. The idea is to use the Chinese remainder theorem: Let $a=\lfloor\log_{10}\frac{N}{k'}\rfloor+1$ and choose pairwise coprime numbers $n_i$ such that $10^{a} ≤ n_n ≤ 10^{a+1}$ and set $\ m_i = N\mod n_i\ $ for $1\leq i\leq k\ (0\leq m_i < n_i)$.
We distribute the $m_i$.
(a) Show that any $k'$ people can use their $m_i$ to find $N$.
(b) How large does $N$ have to be in terms of $k$ and $k'$ such that fewer than $k'$ people cannot obtain the secret?
You can assume that there are plenty of primes in the interval $[10^a , 10^{a+1}]$, roughly $\frac{10^{a+1}}{a}$.
I wrote down a small example with $k=4$ and $k'=2$, but even then I could not figure out how to reconstruct $N$. You need to know the moduli $n_i$ in order to reconstruct $N$ with the Chinese remainder theorem, or not? So how could I get knowledge of them with only my $m_i$'s? The only thing I could think of calculating are the prime factors of the $m_i$'s.
Does someone have a hint for me on how to solve this? Thank you in advance!