Without calculating, tell which is a perfect square: 1022121; 2042122; 3063126; 4083128

Question

I am trying aptitude questions but was struck on this problem.

Which of the following numbers is a perfect square?

A) $1022121\quad$ B) $2042122\quad$ C) $3063126\quad$ D)$4083128$

(original problem image)

As a perfect square always has last digit $0$, $1$, $4$, $5$, $6$, $9$. So, B and D are eliminated.

But I don't know how to eliminate A or C.

The answer is A.

Answer 1

Perfect squares give remainder $0$ or $1$ after division by $3$ and also $4$, both of which can be checked quickly. The C gives same remainder as $26$ after division by $4$, which is $2\neq 0,1$.

Answer 2

Using the method of casting out nines, you can see that $3\mid3\,063\,126$, but $9\nmid3\,063\,126$. So, $3\,063\,126$ cannot be a perfect square.

Answer 3

For the final two digits of a square in base $10$ you have $00$ or $25$ or $E1$ or $E4$ or $[O]6$ or $E9$ where O and E stand for odd and even. You cannot have $E6$.

Just to complete the three methods I spotted, since two have been given by others. This is a modest extension of your existing method.