Question 6
Let $a_1,a_2,a_3,\ldots$ be an infinite sequence of positive integers greater than 1. Suppose that for all positive integers $n$, the number $a_{n+1}$ is the smallest positive integer greater than $a_n$ such that $\gcd(a_{n+1},a_i)\gt 1$ for every $i=1,2,\ldots,n$. Prove that there exist positive integers $T$ and $L$ such that
$$a_{n+T}=a_n+L$$
for every positive integer $n$. Here $\gcd(x,y)$ denotes the greatest common divisor of positive integers $x,y$.
$$a_{n+T}=a_n+L$$
for every positive integer $n$. Here $\gcd(x,y)$ denotes the greatest common divisor of positive integers $x,y$.