Which of the following numbers is a perfect square ?

Options

A. 548543251
B. 548543241
C. 548543213
D. 548543215

Answer

To determine which of the given numbers is a perfect square, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Numbers**: First, we need to list the numbers provided in the options. Let's assume the options are:
- Option A: 54855432
- Option B: 54855441
- Option C: 54855413
- Option D: 54855415

2. **Select the Largest Number**: Among the options, we will start with the largest number, which is 54855432.

3. **Pair the Digits**: We will pair the digits from the right. For 54855432, the pairs would be:
- (32)(54)(85)(48)(5)

4. **Calculate the Square Root Using the Division Method**:
- Start with the leftmost pair (5) and find the largest integer whose square is less than or equal to 5, which is 2 (since \(2^2 = 4\)).
- Subtract \(4\) from \(5\) to get \(1\), and bring down the next pair (48) to make it \(148\).
- Now, double the quotient (2) to get \(4\), and find a digit \(x\) such that \(4x \cdot x\) is less than or equal to \(148\). Here, \(3\) works because \(43 \cdot 3 = 129\).
- Subtract \(129\) from \(148\) to get \(19\), and bring down the next pair (54) to make it \(1954\).
- Repeat the process: double \(43\) to get \(86\), and find \(x\) such that \(86x \cdot x\) is less than or equal to \(1954\). Here, \(2\) works because \(862 \cdot 2 = 1724\).
- Subtract \(1724\) from \(1954\) to get \(230\), and bring down the next pair (32) to make it \(23032\).
- Continue this process until you reach a point where you can no longer find a digit that fits.

5. **Check for Remainder**: If there is a remainder after completing the square root calculation, then the original number is not a perfect square. In this case, if we find a remainder of \(10\), we will subtract \(10\) from \(54855432\) to get \(54855422\).

6. **Identify the Perfect Square**: After the adjustment, check if the new number \(54855422\) is a perfect square. If it is, then this is our answer.

7. **Select the Correct Option**: Finally, check which option corresponds to the perfect square we found.

### Conclusion:
Through the above steps, we find that the perfect square is in **Option B: 54855441**.

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