The product of two consecutive odd numbers is 1763. What is the larger number

Options

A. 43
B. 39
C. 41
D. 37

Answer

To find the larger of two consecutive odd numbers whose product is 1763, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Odd Numbers**:
Let the first odd number be \( x \). Since we are looking for consecutive odd numbers, the next odd number will be \( x + 2 \).

2. **Set Up the Equation**:
According to the problem, the product of these two numbers is 1763. Therefore, we can write the equation:
\[ x(x + 2) = 1763 \]

3. **Expand the Equation**:
Expanding the left-hand side gives us:
\[ x^2 + 2x = 1763 \]

4. **Rearrange the Equation**:
To form a standard quadratic equation, we rearrange it:
\[ x^2 + 2x - 1763 = 0 \]

5. **Factor the Quadratic Equation**:
We need to factor the quadratic equation. We look for two numbers that multiply to \(-1763\) and add up to \(2\). After checking possible factors, we find:
\[ (x + 43)(x - 41) = 0 \]

6. **Solve for \( x \)**:
Setting each factor equal to zero gives us:
\[ x + 43 = 0 \quad \text{or} \quad x - 41 = 0 \]
This results in:
\[ x = -43 \quad \text{or} \quad x = 41 \]

7. **Choose the Positive Odd Number**:
Since we are looking for odd numbers, we discard \( x = -43 \) and take \( x = 41 \).

8. **Find the Larger Number**:
The larger of the two consecutive odd numbers is:
\[ x + 2 = 41 + 2 = 43 \]

### Conclusion:
The larger number is **43**.

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