Question 84
Let $n \geq 2$ be an integer and $p_{1} \lt p_{2} \lt \cdots \lt p_{n}$ prime numbers. Prove that there exists an integer $k$ relatively prime with $p_{1} p_{2} \cdots p_{n}$ and such that $\operatorname{gcd}\left(k + p_{1} p_{2} \cdots p_{i},\ p_{1} p_{2} \cdots p_{n}\right) = 1$ for all $i = 1, 2, \ldots, n-1$.