The product of two consecutive odd numbers is 6723. What is the greater number?

Options

A. 89
B. 85
C. 91
D. 83

Answer

To find the greater of two consecutive odd numbers whose product is 6723, we can follow these steps:

### Step 1: Define the Odd Numbers
Let the two consecutive odd numbers be represented as:
- First odd number: \(2n - 1\)
- Second odd number: \(2n + 1\)

### Step 2: Set Up the Equation
According to the problem, the product of these two numbers is given by:
\[ (2n - 1)(2n + 1) = 6723 \]

### Step 3: Use the Difference of Squares
The expression \((2n - 1)(2n + 1)\) can be simplified using the difference of squares formula:
\[ (2n)^2 - 1^2 = 6723 \]
This simplifies to:
\[ 4n^2 - 1 = 6723 \]

### Step 4: Rearrange the Equation
Now, we can rearrange the equation to isolate \(4n^2\):
\[ 4n^2 = 6723 + 1 \]
\[ 4n^2 = 6724 \]

### Step 5: Solve for \(n^2\)
Next, divide both sides by 4 to find \(n^2\):
\[ n^2 = \frac{6724}{4} \]
Calculating this gives:
\[ n^2 = 1681 \]

### Step 6: Find \(n\)
Now, take the square root of both sides to find \(n\):
\[ n = \sqrt{1681} = 41 \]

### Step 7: Find the Greater Odd Number
Now that we have \(n\), we can find the greater of the two consecutive odd numbers:
\[ \text{Greater odd number} = 2n + 1 = 2 \times 41 + 1 \]
Calculating this gives:
\[ = 82 + 1 = 83 \]

### Final Answer
Thus, the greater of the two consecutive odd numbers is **83**.

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