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

Options

A. 6859
B. 649
C. 13824
D. 42875

Answer

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

### Step-by-Step Solution:

1. **Understanding Even Numbers**:
- An even number can be expressed in the form \(2k\), where \(k\) is an integer (this includes positive integers, negative integers, and zero).

2. **Finding the Cube of an Even Number**:
- If \(x = 2k\), then the cube of \(x\) is:
\[ x^3 = (2k)^3 = 2^3 \cdot k^3 = 8k^3 \]
- This means that the cube of an even number is always a multiple of 8.

3. **Identifying the Form of the Cube**:
- Since \(8k^3\) can be factored as \(2 \cdot (4k^3)\), it shows that the cube of an even number is also in the form of \(2m\) (where \(m\) is an integer), confirming that it is even.

4. **Analyzing the Given Options**:
- We need to check each number to see if it is a cube of an even number. We can do this by checking if the number is a multiple of 8.
- Let's say the options are: 1, 3, 8, 24.

5. **Checking Each Option**:
- **1**: Not a cube of an even number (since \(1 = 1^3\)).
- **3**: Not a cube of an even number (since \(3 = 1.442^3\)).
- **8**: This is \(2^3\), which is the cube of the even number 2.
- **24**: To check if 24 is a perfect cube, we can factor it:
\[ 24 = 2^3 \cdot 3^1 \]
Since the powers of the prime factors are not all multiples of 3, 24 is not a perfect cube.

6. **Conclusion**:
- The only number among the options that is the cube of an even number is **8**.

### Final Answer:
The cube of an even number from the options given is **8**.