TITLE: Is the number 100k+11 ever a square?
QUESTION [37 upvotes]: Is the number 100k+11 ever a square?
Obviously all such numbers end in 11. Can a square end with an 11?

REPLY [3 votes]: A simple solution
$$(10x+y)^2 = 100x^2 + 20 xy + y^2$$
where $x$ is any number, and $0 \le y \le 9$

 The first term on right hand side must end in two zeros.
 The second term is a multiple of 20 and therefore must end in 00, 20, 40, 60, or 80
 For the square to end in 1 $y$ must be either 1 or 9
 In which case $y^2$ is either 01 or 81
 Thus 4. and 2. imply that the square must end in either 01, 21, 41, 61, or 81