Primes classifications [closed]

Question

1) If $p_1$$p_2$,...,$p_k$ be different primes and m = product of primes $p_1$,$p_2$,...,$p_k$ . How to prove that, when N = $N_1$ + $N_2$+...+$N_k$, where the prime factors of $N_i$ (here i is running from 1 to k) are exactly {$p_1$,$p_2$,...,$p_k$} \ {$p_i$} for each i, then (N, m) = 1.

2) For any given integer N with (N, m) = 1, use the Chinese Remainder Theorem to determine integers $M_i$ for which N is congruent to $M_i$ (mod m), where i is from 1 to k. The prime factors of $M_i$ are exactly {$p_1$,$p_2$,...,$p_k$} \ {$p_i$} and 1 < or equal to $M_i$ less than or equal to m,for each i.

I need complete discussion on both questions by proof or generalization.

Answer 1

For 1, consider that the only prime factors of $m$ are the $p_i$. For each $N_i$, factor $p_i$ out of the other summands in $N$ and reduce it modulo $p_i$ to conclude that $p_i$ does not divide $N$ for any $i$.

For 2, use the $N_i$ from 1.