Which among the following is a perfect square
The Correct Answer and Explanation is:
The correct answer is (B) 323761.
Explanation:
A perfect square is a number that results from squaring a whole number. To determine whether a number is a perfect square, we can begin by estimating its square root and then checking whether the square of that root gives the original number. Alternatively, we can apply divisibility rules and digital root techniques to eliminate candidates quickly before verifying with exact calculations.
Let’s examine 323761:
The square root of 323761 is approximately 569. Calculating directly:
5692=569×569=(500+69)2=5002+2×500×69+692=250000+69000+4761=323761569^2 = 569 \times 569 = (500 + 69)^2 = 500^2 + 2 \times 500 \times 69 + 69^2 = 250000 + 69000 + 4761 = 323761
This confirms that 323761 is indeed a perfect square because it is the square of the integer 569.
Now, consider the digital root method. If a number is a perfect square, its digital root must be 1, 4, 7, or 9.
- 323761 → 3+2+3+7+6+1 = 22 → 2+2 = 4, which is acceptable for a perfect square.
- For comparison:
- 455562 → digits sum to 27 → 2+7 = 9 (possible but needs checking)
- 675487 → digits sum to 37 → 3+7 = 10 → 1+0 = 1 (possible)
- 897568 → digits sum to 43 → 4+3 = 7 (possible)
- 347860 → digits sum to 28 → 2+8 = 10 → 1+0 = 1 (possible)
While many options have valid digital roots, only 323761 produces an integer when square-rooted exactly. All others result in non-integer values.
Hence, 323761 is a perfect square and the correct choice.
