jmc

An integer is divisible by $18$ if and only if the sum of its digits is divisible by $9$ and the last digit is even (meaning it is divisible by both 9 and 2). The sum of the digits of $-11213141$ is 14. Since $-11213141$ is negative, this number is 5 $\text{less than}$ a multiple of 9. The number is 4 $\text{more than}$ a multiple of 9. Subtracting 4 gives $$-11213141=-11213145+4.$$Since $-11213145$ has a digit sum of 18, this number is a multiple of 9. However this is not a multiple of 18 so we need to subtract 9 again: $$-11213141=-11213154+13.$$Now the number $-11213154$ is a multiple of 18, so the answer is $\boxed{13}$. $$-11213141\equiv 13(\mathrm{mod}18).$$