congrats to @mathfortress and @happychau123 for getting the correct answer!

My solution is the same as @happychau123 's and this method does not require calculator at all! Great job!

So a short summary:
A number is divisible by 12 if it is divisible by 3 and 4.
Method of checking if a number is divisible by 3: Sum up all its digits and see if it is divisible by 3.
Method of checking if a number is divisible by 4: See if the number formed by the last 2 digits are divisible by 4.
Therefore, the closest number is 111...1116.
Because the sum of digits is 2022, which is divisible by 3; and its final 2 digits is 16, which is divisible by 4.
Therefore the remainder is 7.