Question 54
Let $m, n$ be integers greater than $1$, and let $a_1, a_2, \dots, a_m$ be positive integers not greater than $n^m$. Prove that there exist positive integers $b_1, b_2, \dots, b_m$ not greater than $n$ such that
$$
\text{gcd}(a_1 + b_1, a_2 + b_2, a_3 + b_3, \dots, a_m + b_m) \lt n,
$$
where $\text{gcd}(x_1, x_2, \dots, x_m)$ denotes the greatest common divisor of $x_1, x_2, \dots, x_m$.
$$
\text{gcd}(a_1 + b_1, a_2 + b_2, a_3 + b_3, \dots, a_m + b_m) \lt n,
$$
where $\text{gcd}(x_1, x_2, \dots, x_m)$ denotes the greatest common divisor of $x_1, x_2, \dots, x_m$.