Proving that sequences with the pattern aaaaaaa don't include a perfect square I need help with the following problem:

Which of the following sequences doesn't have perfect square:
A) $11, 111, 1111, \dots$
B) $33, 333, 3333, \dots$
C) $44, 444, 4444, \dots$
D) $77, 777, 7777, \dots$

I proved that the first sequence cannot contain a perfect square, but it seems that none of them contains a perfect square. Is it possible to  write a proof for a general pattern $aaaaaaaaa$?

For 1 you are done.
For 2 not divisible by 4.
For 3 no perfect square ends with digit 3.
For 4 just divide by 4 and get back to the case with 1.
For 5 not divisible by 25.
For 6 not divisible by 4.
For 7 no perfect square ends in digit 7.
For 8 not divisible by 16.
For 9 divide by 9 and get back to 1.
