Question 117
Find the largest integer $k$ ($k \geq 2$), for which there exists an integer $n$ ($n \geq k$) such that from any collection of $n$ consecutive positive integers one can always choose $k$ numbers, which verify the following conditions:
1. each chosen number is not divisible by $6$, by $7$ and by $8$;
2. the positive difference of any two different chosen numbers is not divisible by at least one of the numbers $6$, $7$ or $8$.
1. each chosen number is not divisible by $6$, by $7$ and by $8$;
2. the positive difference of any two different chosen numbers is not divisible by at least one of the numbers $6$, $7$ or $8$.