MathDB

Find the largest integer

k

(

k \ge 2

), for which there exists an integer

n

(

n \ge 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

, nor by

8

; 2. the positive difference of any two distinct chosen numbers is not divisible by at least one of the numbers

6

7

, and

8

Problem Statement