Marinada '25

[lang=en] Find the smallest positive integer $n$, such that for any integers $a_1,a_2,\ldots ,a_n; b_1,b_2,\ldots ,b_n$ there exists integers $x_1,x_2,\ldots ,x_n$ satisfying the following two conditions: i) There exists $i\in \{1,2,\ldots ,n\}$ such that $x_i$ and $2024$ are coprime ii) $\sum^n_{i=1} a_ix_i \equiv \sum^n_{i=1} b_ix_i \equiv 0 \pmod {2024}$ [/lang] [lang=hr] Nađi najmanji cijeli broj $n$ tako da za bilo koje cijele brojeve $a_1,a_2,\ldots ,a_n; b_1,b_2,\ldots ,b_n$ postoje cijeli brojevi $x_1,x_2,\ldots ,x_n$ koji zadovoljavaju sljedeća dva uvjeta : 1) Postoji $i\in \{1,2,\ldots ,n\}$ tako da $x_i$ i $2024$ su relativno prosti. 2) $\sum^n_{i=1} a_ix_i \equiv \sum^n_{i=1} b_ix_i \equiv 0 \pmod {2024}$ [/lang]