jmc

List the first few multiples of 9: $9,18,27,36,\dots$. We see that these are all lucky integers because their digits sum to 9, and the pattern of increasing the first digit by 1 while decreasing the second digit by 1 preserves this property. However, this pattern stops after the last digit reaches zero. Indeed, 90 is still a lucky integer, but 99 is not, since the digits sum to 18 and 99 is not divisible by 18. Thus $\boxed{99}$ is the least positive multiple of 9 which is not a lucky integer.