Which of the following numbers could be a perfect square?

The correct answer is 273529.

To determine which of the given numbers could be a perfect square, we can evaluate each option using the properties of perfect squares:

1. Unit Digit Property:
The square of any integer can only end in the digits 0, 1, 4, 5, 6, or 9. A perfect square can never end in 2, 3, 7, or 8.
For 415898, the units digit is 8. Thus, 415898 cannot be a perfect square.

2. Tens Digit Rule for Numbers Ending in 5:
If a perfect square ends in 5, the tens digit must always be 2 (so the number must end in 25).
For 384135, the last two digits are 35. Since the tens digit is 3 (not 2), 384135 cannot be a perfect square.

3. Tens Digit Rule for Numbers Ending in 6:
If a perfect square ends in 6, the tens digit must always be odd (1, 3, 5, 7, or 9).
For 632146, the last two digits are 46. Since the tens digit is 4 (which is even), 632146 cannot be a perfect square.

4. Analyzing 273529:
The number 273529 ends in 29 (units digit 9, tens digit 2), which satisfies the conditions for a perfect square.
We can verify this directly:

${523}^2=523\times 523=273529$

Therefore, 273529 is a perfect square.