There are two-digit perfect squares whose square roots are the same as their units digits. What is the sum of these squares?
25+36
To find the two-digit perfect squares whose square roots are the same as their units digits, we need to systematically calculate the square of each two-digit number and check if its units digit is equal to its square root.
Let's start by listing all the two-digit numbers:
10, 11, 12, ..., 19,
20, 21, 22, ..., 29,
...
90, 91, 92, ..., 99.
Now, let's calculate the square of each number:
For 10, the square is 10^2 = 100. The units digit is 0, which is not equal to 10, so it is not a valid square.
For 11, the square is 11^2 = 121. The units digit is 1, which is equal to 11, so it is a valid square.
For 12, the square is 12^2 = 144. The units digit is 4, which is not equal to 12, so it is not a valid square.
...
For 19, the square is 19^2 = 361. The units digit is 1, which is equal to 19, so it is a valid square.
For 20, the square is 20^2 = 400. The units digit is 0, which is not equal to 20, so it is not a valid square.
If we continue this process for all the two-digit numbers, we find that the two-digit perfect squares whose square roots are equal to their units digits are:
11, 22, 33, 44, 55, 66, 77, 88, 99.
Now we can calculate their sum:
11 + 22 + 33 + 44 + 55 + 66 + 77 + 88 + 99 = 495.
So, the sum of these squares is 495.