Question 98
Let $n$ be a strictly positive integer and let $a, a_{1}, \ldots, a_{n}$ be strictly positive integers. Suppose that for every integer $k$ for which the integer $a k+1$ is a perfect square, at least one of the integers $a_{1} k+1, \ldots, a_{n} k+1$ is also a perfect square.
Prove that there exists an index $1 \leqslant i \leqslant n$ such that $a=a_{i}$.
Prove that there exists an index $1 \leqslant i \leqslant n$ such that $a=a_{i}$.