There is unique 3 digit number which is cube of a natural number, if we shift the position of the digits of this number. The new number also becomes the cube of another number. The number is :

Options

A. 343
B. 729
C. 125
D. does not exist

Answer

To solve the problem, we need to find a unique three-digit number that is the cube of a natural number, and when we rearrange its digits, it also becomes the cube of another natural number.

### Step-by-Step Solution:

1. **Identify the range of cubes**:
- The smallest cube of a natural number that is a three-digit number is \( 5^3 = 125 \).
- The largest cube of a natural number that is a three-digit number is \( 9^3 = 729 \).
- Therefore, we need to consider the cubes of the numbers 5, 6, 7, 8, and 9.

2. **List the cubes of natural numbers from 5 to 9**:
- \( 5^3 = 125 \)
- \( 6^3 = 216 \)
- \( 7^3 = 343 \)
- \( 8^3 = 512 \)
- \( 9^3 = 729 \)

3. **Check each cube to see if its digits can be rearranged to form another cube**:
- **For \( 125 \)**:
- Rearranging gives: 215, 251, 512.
- Check if any of these are cubes: \( 512 = 8^3 \) (valid).
- **For \( 216 \)**:
- Rearranging gives: 126, 162, 621, 612.
- None of these are cubes.
- **For \( 343 \)**:
- Rearranging gives: 334, 433.
- None of these are cubes.
- **For \( 512 \)**:
- Rearranging gives: 125, 152, 215, 251, 521.
- \( 125 = 5^3 \) (valid).
- **For \( 729 \)**:
- Rearranging gives: 297, 279, 972, 927, 792.
- None of these are cubes.

4. **Conclusion**:
- The only number that satisfies the condition is \( 512 \), which is \( 8^3 \), and its rearrangement \( 125 \) is \( 5^3 \).

### Final Answer:
The unique three-digit number is **512**.