Maximum subset which has no sum of two divisible by K

Question

I am given the set {1, 2, 3, ... ,N}. I have to find the maximum size of a subset of the given set so that the sum of any 2 numbers from the subset is not divisible by a given n

Answer 1

formula is

|N/K| * |(K-1)/2| + ost 

ost =
if n

where |a/b| for example |9/2| = 4 |7/2| = 3

example n = 30 , k =7 ;

1 2 3 4 5 6 7
8 9 10 11 12 13 14
15 16 17 18 19 20 21
22 23 24 25 26 27 28
29 30

1 2 3 |4| 5 6 7. - is first line . 8 9 10 |11| 12 13 14 - second line if we getting first 3 number in each line we may get size of this subset. also we may adding one number from ( 7 14 28)

getting first 3 number (1 2 3) is a number |(k-1)/2| . a number of this line is |n/k| . if there is not residue we may add one number (for example last number). if residue < |(k-1)/2| we get all number in last line else getting |(K-1)/2|.

thanks for exception case. ost = 0 if k>n