What is that least digit that must be added to the product `5786 xx 5784` to make it a perfect square?

Options

A. 1
B. 6
C. 5
D. 4

Answer

To find the least digit that must be added to the product \( 5786 \times 5784 \) to make it a perfect square, we can follow these steps:

### Step 1: Calculate the Product
First, we need to calculate the product of \( 5786 \) and \( 5784 \).

\[ 5786 \times 5784 \]

### Step 2: Identify the Nearest Perfect Square
Next, we need to find the nearest perfect square to the product we calculated in Step 1. We can do this by finding the square root of the product and rounding it to the nearest whole number.

Let’s denote the product as \( P \):
\[ P = 5786 \times 5784 \]
Now, calculate the square root of \( P \):
\[ \sqrt{P} \]

### Step 3: Round to the Nearest Whole Number
Round the result from Step 2 to the nearest whole number. Let’s call this number \( n \).

### Step 4: Calculate \( n^2 \)
Now, calculate \( n^2 \):
\[ n^2 \]

### Step 5: Determine the Difference
Next, we need to find the difference between \( n^2 \) and \( P \):
\[ \text{Difference} = n^2 - P \]

### Step 6: Check if the Difference is a Digit
If the difference is a single-digit number (0-9), that is the least digit that must be added to \( P \) to make it a perfect square. If the difference is greater than 9, we need to find the least digit that can be added to make it a perfect square.

### Step 7: Conclusion
The least digit \( x \) that must be added to \( P \) to make it a perfect square is the result from Step 6.

### Final Answer
After performing all the calculations, we find that the least digit \( x \) that must be added is \( 1 \).

---