Question 29
Let $p$ be a prime number. Determine the largest possible $n$ such that the following holds. It is possible to fill an $n \times n$ table with integers $a_{ik}$ in the $i$-th row and $k$-th column, for $1 \le i, k \le n$, such that for any quadruple $i, j, k, l$ with $1 \le i \lt j \le n$ and $1 \le k \lt l \le n$, the number $a_{ik}a_{jl} - a_{il}a_{jk}$ is not divisible by $p$.