Among the integers 1, 2,3, 4,……, 2018, what is the maximum number of integers that can be selected such that the sum of any two selected numbers is not a multiple of 7?

In the set{1,2,3,4,5,…..,2018}, there are 289 numbers of the form ‘7k +1’

(1,8,15,….., 2017),

289 numbers of the form ‘7k+2’

(2, 9 , 16 , ….., 2018)

288 numbers of the form ‘7k + 3’

(3 , 10 , 17 , ……2012)

We cannot take a number of the form ‘7k+ 4’, ‘7k+5’ and ‘7k+ 6’ as they are the inverse of ‘7k +3’, ‘7k+52’ and ‘7k + 1’. Now, we can take exactly one number of the form 7k. Therefore, we can take a maximum of 289 + 289 + 288 + 1 = 867 integers.