state true or false : Is 246 a perfect cube?

Answer

To determine whether 246 is a perfect cube, we need to analyze its prime factorization and check the powers of the prime factors.

### Step-by-Step Solution:

1. **Find the Prime Factorization of 246**:
- Start by dividing 246 by the smallest prime number, which is 2.
- \( 246 \div 2 = 123 \)
- So, we have \( 246 = 2 \times 123 \).

2. **Factor 123**:
- Next, we factor 123. The smallest prime number that divides 123 is 3.
- \( 123 \div 3 = 41 \)
- Thus, we have \( 123 = 3 \times 41 \).
- Therefore, the complete factorization of 246 is \( 246 = 2 \times 3 \times 41 \).

3. **Identify the Powers of the Prime Factors**:
- From the factorization \( 246 = 2^1 \times 3^1 \times 41^1 \), we see that the powers of the prime factors are:
- Power of 2: 1
- Power of 3: 1
- Power of 41: 1

4. **Check if the Powers are Multiples of 3**:
- For a number to be a perfect cube, all the powers in its prime factorization must be multiples of 3.
- Here, the powers are 1, 1, and 1, which are not multiples of 3.

5. **Conclusion**:
- Since not all the powers of the prime factors of 246 are multiples of 3, we conclude that 246 is **not a perfect cube**.

### Final Answer:
**False**: 246 is not a perfect cube.

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