Question 117
a) Consider $n$ coprime natural numbers greater than $1$ like $d_1, d_2, \dots, d_n$ and arbitrary natural numbers $r_1, r_2, \dots, r_n$. Prove that there exists a natural number $x$, $1 \le x \le 3^n$, that satisfies the following system of modular inequalities:
$$
\begin{array}{l}
x \not\equiv r_1 \pmod{d_1} \\
x \not\equiv r_2 \pmod{d_2} \\
\vdots \\
x \not\equiv r_n \pmod{d_n}
\end{array}
$$
b) For each real number $\epsilon \gt 0$, prove that there exists a number $N$ such that for each natural number $n \gt N$ and natural numbers $d_1, d_2, \dots, d_n$ and $r_1, r_2, \dots, r_n$, where $d_i$'s ($1 \le i \le n$) are coprime, the above system of inequalities has a solution $x$ that $1 \le x \le (2 + \epsilon)^n$.
$$
\begin{array}{l}
x \not\equiv r_1 \pmod{d_1} \\
x \not\equiv r_2 \pmod{d_2} \\
\vdots \\
x \not\equiv r_n \pmod{d_n}
\end{array}
$$
b) For each real number $\epsilon \gt 0$, prove that there exists a number $N$ such that for each natural number $n \gt N$ and natural numbers $d_1, d_2, \dots, d_n$ and $r_1, r_2, \dots, r_n$, where $d_i$'s ($1 \le i \le n$) are coprime, the above system of inequalities has a solution $x$ that $1 \le x \le (2 + \epsilon)^n$.