What is the car no. and how to derive it logically - see details below?

Question

A statistician and a mathematician saw a car number and couldn't recall it later.
But the former did remember that the first two digits were identical just as the last two digits too were repetitive. The mathematician could recall that he had noted the number to be a whole square. And then it took him seconds to work out the car number without any doubt whatsoever.
You are the mathematician - please show how to work out.

Answer 1

Let the digits be 'a' and 'b'.
Now the number is "aabb".
The value of the number is: b + 10*b + 100*a + 1000* a
= 11*b + 1100*a = 11(b+100*a).

Now, this is a perfect square. Therefore 100*a + b has to be a multiple of 11.

Since a and b are digits, (and a can't be 0) the number looks like "a0b". If this has to be a multiple of 11, a+b has to be 11.

combinations, "a0b" and "aabb" values:
2 and 9, 9 and 2; (209 or 902); (2299 or 9922)
3 and 8, 8 and 3.(308 or 802); (3388 or 8833)
4 and 7, 7 and 4.(407 or 704); (4477 or 7744)
5 and 6, 6 and 5.(506 or 605); (5566 or 6655)

Only one is a whole square-> 7744. That is your answer.

Answer 2

Guess and check

7744

which is the square of 88

Answer 3

Try 88 square it is 7744.
Did this manually so i cant show i got it.