The sum is?

The possible numbers are 111,222 and so on up to 999. See that all these numbers are multiples of 111 and the factors of 111 are 37 and 3.So n or n+1 should be a multiple of 37 or the number itself. So the possible values of n and n+1 are 36,37 or 37,38.Taking n =36 gives the answer.

The three digit number we are looking for is 666. now we know that the sum of first 10 positive integers is 55 (1 to 10). next, from 11 to 20. the sum will be 100+55. for 21-30, the sum will be 200+55. sum of 1-30 is 465. now we keep adding each number from 31 till we get all digits of the total similar. this gives us 36. hence 3+6=9.