The number 55 is attached to the right of a two-digit number, and the resulting 4-digit number is divisible by 21. what could the 2-digit number be?

Your 2-digit number could be any of {11, 32, 53, 74, 95}.
  1155 = 21·55
  3255 = 21·155
  5355 = 21·255
  7455 = 21·355
  9555 = 21·455


The smallest possible 4-digit number of interest is 1055. It has a remainder of 5 when divided by 21. Adding 100 to the number to make it be 1155 adds 100 mod 21 = 16 to the remainder, making it be 21 (or zero). Thus 1155 is the smallest 4-digit number of interest, and 11 is the corresponding 2-digit number. Of course, adding a multiple of 2100 to the 4-digit number will ensure that it remains divisible by 21. So, your 2-digit numbers are 21n+11 for n=0 to 4.