What is that least number that must be added to the product `684 xx 686` to make it a perfect square?

Options

A. 685
B. 1
C. 684
D. 686

Answer

To find the least number that must be added to the product \( 684 \times 686 \) to make it a perfect square, we can follow these steps:

### Step 1: Calculate the product
First, we need to calculate the product \( 684 \times 686 \).

\[ 684 \times 686 = 684^2 - 2^2 \quad \text{(using the difference of squares)} \]

### Step 2: Simplify the expression
Now, we can simplify \( 684^2 - 2^2 \):

\[ 684^2 - 4 \]

### Step 3: Identify the nearest perfect square
Next, we need to find the nearest perfect square greater than \( 684^2 - 4 \). The perfect square we are looking for is of the form \( n^2 \).

### Step 4: Calculate \( 684^2 \)
Calculate \( 684^2 \):

\[ 684^2 = 467056 \]

Thus,

\[ 684^2 - 4 = 467056 - 4 = 467052 \]

### Step 5: Find the next perfect square
Now, we need to find the smallest integer \( n \) such that \( n^2 > 467052 \).

Calculating the square root:

\[ \sqrt{467052} \approx 684.3 \]

The next integer is \( 685 \). Thus, we calculate \( 685^2 \):

\[ 685^2 = 469225 \]

### Step 6: Calculate the difference
Now, we find the difference between \( 685^2 \) and \( 467052 \):

\[ 469225 - 467052 = 2173 \]

### Step 7: Conclusion
Thus, the least number that must be added to \( 684 \times 686 \) to make it a perfect square is:

\[ \boxed{2173} \]

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