Find the only four-digit perfect square that is a pair of two-digit squares written side by side?
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To find the only four-digit perfect square that is a pair of two-digit squares written side by side, we need to break down the problem.
First, let's consider the four-digit perfect squares. The least four-digit perfect square is 100^2 = 10,000, and the greatest four-digit perfect square is 31^2 = 9,681. So we need to find a number between these two limits that satisfies the given conditions.
Next, let's find the two-digit squares. The two-digit squares range from 10^2 = 100 to 99^2 = 9,801. We can see that the two-digit squares do overlap with the range of four-digit perfect squares.
Now, we need to find a pair of two-digit squares written side by side that forms a four-digit perfect square. We can start by listing the squares from 10^2 to 31^2, and combine them to check if any of them form a four-digit square.
10^2 = 100
11^2 = 121
12^2 = 144
...
30^2 = 900
31^2 = 961
Combining any two of these squares, we get the following possibilities for four-digit numbers:
100121, 100144, 100169, ..., 900784, 900900, 900961
110044, 110121, 110144, ..., 901801, 901900, 901961
...
961196
Out of these possibilities, only one number is a perfect square: 8281 = 91^2.
Therefore, the only four-digit perfect square that is a pair of two-digit squares written side by side is 8281.