Can a number like ABAB, where A and B are numbers, be an exact square?

Consider the number ABAB.

For any A and B, the number ABAB is the product of AB * 101.

The full square of any number, by definition, is the product of that number by itself.

The number 101 is a prime number, that is, it cannot be decomposed into any factors, therefore, the product of number 101 by another number can be a perfect square only if the second factor is also 101. But by the condition of the problem, the second factor AB is a two-digit number, therefore the number ABAB is not can be a perfect square of any number.