How to prove mutually independence?

Question

The original problem is somewhat trivial, and I have one last step to prove the mutually independence.

$P(Y=k)=1/n, k=0,1,...,n-1$

$n=p_1p_2…p_k$ and $p_1,p_2,...,p_k$ are co-prime integers.

Prove that $X_{p_i}=(Y\ mod\ p_i), i=1,...,k$ are mutually independent.

Any help would be appreciated.

(By the way, I have caculated the probability $P(X_{p_i}=k)=1/p_i, k=1,...,p_i-1$

But no idea of what to do next.)

Answer 1

Here is my solution to it, as a reference.