Find the least number which must be added to `5483` so that the resulting number is a perfect square.

Answer

To find the least number that must be added to 5483 so that the resulting number is a perfect square, we can follow these steps:

### Step 1: Find the square root of 5483
We will use the long division method to find the square root of 5483.

### Step 2: Pair the digits
Starting from the right, we will pair the digits of 5483. We have:
- Pair 1: 83
- Pair 2: 54

### Step 3: Find the largest integer whose square is less than or equal to the first pair (54)
The largest integer whose square is less than 54 is 7 (since \(7^2 = 49\)).

### Step 4: Subtract and bring down the next pair
Subtract \(49\) from \(54\):
\[ 54 - 49 = 5 \]
Now, bring down the next pair (83) to make it 583.

### Step 5: Double the quotient and find the next digit
Double the quotient (7) to get 14. Now we need to find a digit \(x\) such that:
\[ (14x) \times x < 583 \]
Testing \(x = 4\):
\[ (144) \times 4 = 576 < 583 \]
Now, we can use \(x = 4\).

### Step 6: Subtract and find the new remainder
Subtract \(576\) from \(583\):
\[ 583 - 576 = 7 \]
Now, we have a remainder of 7.

### Step 7: Identify the next perfect square
Since we have determined that \(74^2 = 5476\) and \(75^2 = 5625\), we can see that 5483 is greater than \(74^2\) but less than \(75^2\).

### Step 8: Calculate the difference to find the least number to add
Now, we need to find out how much we need to add to 5483 to reach the next perfect square, which is \(5625\):
\[ 5625 - 5483 = 142 \]

### Conclusion
The least number that must be added to 5483 to make it a perfect square is **142**.

---