Is the number 100k+11 ever a square? Is the number 100k+11 ever a square?
Obviously all such numbers end in 11. Can a square end with an 11?

Ends by 11 means, $n² \mod 100 = 11$ which also implies end by one $n² \mod 10 = 1$ 
this is true if and only if $(n \mod 100)^2 \mod 100 = 11$ and $(n \mod 10)² \mod 10 = 1$. 
Ergo you juste have to check squares of 1, 11, 21, 31, 41, 51, 61, 71, 81 and 91 and see none ends with 11.
