Which of the following numbers is the cube of an odd number?

Options

A. 729
B. 2744
C. 32768
D. 1728

Answer

To determine which of the following numbers is the cube of an odd number, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Odd Numbers**:
An odd number can be expressed in the form of \(2k + 1\), where \(k\) is any integer.

2. **Finding the Cube of an Odd Number**:
To find the cube of an odd number, we take \((2k + 1)^3\).

3. **Expanding the Cube**:
Using the binomial expansion, we can expand \((2k + 1)^3\):
\[ (2k + 1)^3 = (2k + 1)(2k + 1)(2k + 1) \]
- First, calculate \((2k + 1)(2k + 1)\):
\[ = 4k^2 + 4k + 1 \]
- Now multiply this result by \((2k + 1)\):
\[ (4k^2 + 4k + 1)(2k + 1) = 8k^3 + 12k^2 + 6k + 1 \]

4. **Identifying the Result**:
The result \(8k^3 + 12k^2 + 6k + 1\) is always odd because it ends with \(+1\). This means that the cube of any odd number is always odd.

5. **Checking Given Numbers**:
Now, we need to check the numbers provided in the question to see which one is odd and a cube:
- For example, if one of the options is 729:
- Check if \(729\) is a cube:
\[ 9 \times 9 = 81 \quad \text{and} \quad 81 \times 9 = 729 \]
- Since \(9\) is an odd number, \(729\) is indeed the cube of an odd number.

6. **Conclusion**:
Therefore, the number that is the cube of an odd number from the options provided is \(729\).