5. Find the smallest positive integer $n$ such that among any $n$ chosen integers, there exist at least two integers whose sum or difference is divisible by 1997.
5. Take 999 integers $0,1,2, \cdots, 998$, let $a_{i}$ and $a_{j}$ be two of them, when $i \neq j$, $a_{i}+a_{j} \leqslant 1995$, and $0999$, so there are at least two different $a_{i}$ and $a_{1}$, such that $a_{i} \equiv -a_{j}(\bmod 1997)$, thus $1997 \mid (a_{i}+a_{j})$, or $a_{i} \equiv a_{j}(\bmod 1997)$, thus $1997 \mid (a_{i}-a_{j})$, hence the suitable $n=1000$.