There is a list with $K$ distinct positive integers so that
- exactly 1 number in the list is divisible by $K$,
- exactly 2 numbers in the list are divisible by $K-1$,
- exactly 3 numbers in the list are divisible by $K-2$,
- ... ... ... ... ...
- exactly $K$ numbers in the list are divisible by $1$.
What's the largest possible value of $K$ under these circumstances?