I think the answer is
9, and possible numbers are 1, 11, 21, 31, 41, 51, 61, 71, 81. If you take the sum of $1 \le n \le 9$ numbers, the last digit of the sum is $n$ so the sum is not divisible by 10.
Glorfindel
- 28.2k
- 9
- 98
- 143