What is that least number that must be addeed 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: Rewrite the Product
We start with the expression \( 684 \times 686 \). Notice that \( 686 \) can be rewritten as \( 684 + 2 \). Thus, we can express the product as:
\[ 684 \times 686 = 684 \times (684 + 2) \]

### Step 2: Expand the Expression
Using the distributive property, we expand the product:
\[ 684 \times (684 + 2) = 684^2 + 2 \times 684 \]

### Step 3: Recognize the Perfect Square Form
The expression \( 684^2 + 2 \times 684 \) can be recognized as part of the expansion of a perfect square:
\[ a^2 + 2ab = (a + b)^2 \]
Here, if we let \( a = 684 \) and \( b = 1 \), we have:
\[ 684^2 + 2 \times 684 \times 1 \]
This means we can express it as:
\[ (684 + 1)^2 - 1^2 = (685)^2 - 1 \]

### Step 4: Determine What to Add
To make \( 684^2 + 2 \times 684 \) a perfect square, we need to add \( 1^2 \) (which is 1) to the expression \( (685)^2 - 1 \):
\[ 684^2 + 2 \times 684 + 1 = (685)^2 \]
Thus, we need to add \( 1 \) to the original product \( 684 \times 686 \) to make it a perfect square.

### Conclusion
The least number that must be added to \( 684 \times 686 \) to make it a perfect square is:
\[ \boxed{1} \]