Which of the following numbers is a perfect cube?

Options

A. 1152
B. 1331
C. 2016
D. 739

Answer

To determine which of the given numbers is a perfect cube, we will analyze each option by finding their prime factors and checking if they can be grouped into sets of three.

### Step-by-Step Solution:

1. **Identify the Options**: The options given are:
- Option A: 1152
- Option B: 1331
- Option C: 2016
- Option D: 739

2. **Check Option A: 1152**
- Factor 1152:
- 1152 ÷ 2 = 576
- 576 ÷ 2 = 288
- 288 ÷ 2 = 144
- 144 ÷ 2 = 72
- 72 ÷ 2 = 36
- 36 ÷ 2 = 18
- 18 ÷ 2 = 9
- 9 ÷ 3 = 3
- 3 ÷ 3 = 1
- Prime factorization: \( 1152 = 2^7 \times 3^2 \)
- Check for perfect cube:
- The powers of 2 (7) and 3 (2) cannot be grouped into sets of three.
- Conclusion: **Not a perfect cube.**

3. **Check Option B: 1331**
- Factor 1331:
- 1331 ÷ 11 = 121
- 121 ÷ 11 = 11
- 11 ÷ 11 = 1
- Prime factorization: \( 1331 = 11^3 \)
- Check for perfect cube:
- The power of 11 (3) can be grouped into one set of three.
- Conclusion: **This is a perfect cube.**

4. **Check Option C: 2016**
- Factor 2016:
- 2016 ÷ 2 = 1008
- 1008 ÷ 2 = 504
- 504 ÷ 2 = 252
- 252 ÷ 2 = 126
- 126 ÷ 2 = 63
- 63 ÷ 3 = 21
- 21 ÷ 3 = 7
- 7 ÷ 7 = 1
- Prime factorization: \( 2016 = 2^5 \times 3^2 \times 7^1 \)
- Check for perfect cube:
- The powers of 2 (5), 3 (2), and 7 (1) cannot be grouped into sets of three.
- Conclusion: **Not a perfect cube.**

5. **Check Option D: 739**
- Factor 739:
- 739 is a prime number.
- Prime factorization: \( 739 = 1 \times 739 \)
- Check for perfect cube:
- The only factors are 1 and 739, which cannot form a perfect cube.
- Conclusion: **Not a perfect cube.**

### Final Answer:
The only number that is a perfect cube is **Option B: 1331**.