Find two consecutive odd natural numbers whose product is 323.

Answer

To find two consecutive odd natural numbers whose product is 323, we can follow these steps:

### Step 1: Define the Variables
Let the first odd natural number be \( n \). Since we are looking for consecutive odd natural numbers, the next odd natural number will be \( n + 2 \).

### Step 2: Set Up the Equation
According to the problem, the product of these two numbers is 323. Therefore, we can write the equation:
\[ n(n + 2) = 323 \]

### Step 3: Expand the Equation
Expanding the left side of the equation gives us:
\[ n^2 + 2n = 323 \]

### Step 4: Rearrange the Equation
To form a standard quadratic equation, we rearrange it as follows:
\[ n^2 + 2n - 323 = 0 \]

### Step 5: Factor the Quadratic Equation
Next, we need to factor the quadratic equation. We are looking for two numbers that multiply to \(-323\) and add to \(2\). The numbers \(19\) and \(-17\) fit this requirement:
\[ (n + 19)(n - 17) = 0 \]

### Step 6: Solve for \( n \)
Setting each factor to zero gives us:
1. \( n + 19 = 0 \) → \( n = -19 \) (not a natural number)
2. \( n - 17 = 0 \) → \( n = 17 \)

Since we are looking for natural numbers, we take \( n = 17 \).

### Step 7: Find the Two Odd Natural Numbers
Now we can find the two consecutive odd natural numbers:
- First number: \( n = 17 \)
- Second number: \( n + 2 = 19 \)

### Conclusion
The two consecutive odd natural numbers whose product is 323 are **17 and 19**.

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