On the book "Quantum computation and quantum information" from Nielsen and Chuang on page 634 at the beginning of a theorem's proof it's assumed to be able to choose a number $x$ uniformly at random from $Z_N^*$ with $N = p_1^{\alpha_1}...p_m^{\alpha_m}$ and $p_j$ a prime factor.
However the book says that this operation is equivalent to choosing $x_j$ independently and uniformly at random from $Z_{p_j^{\alpha_j}}^*$ and requiring that $x = x_j\pmod{p_j^{\alpha_j}}$ for each $j$.
I understood that this system has a solution thanks to the Chinese remainder theorem but how is the solution $x$ co-prime to $N$?