Find which of the following are perfect cubes ?
24000

Answer

To determine whether 24000 is a perfect cube, we will follow these steps:

### Step 1: Prime Factorization
We start by factorizing 24000 into its prime factors.

1. Since 24000 is an even number, we can divide it by 2:
- \( 24000 \div 2 = 12000 \)
2. Continuing to divide by 2:
- \( 12000 \div 2 = 6000 \)
- \( 6000 \div 2 = 3000 \)
- \( 3000 \div 2 = 1500 \)
- \( 1500 \div 2 = 750 \)
3. Now, 750 is not even, so we check for divisibility by 3 (sum of digits is 7+5+0=12, which is divisible by 3):
- \( 750 \div 3 = 250 \)
4. Next, we factor 250:
- \( 250 \div 5 = 50 \)
- \( 50 \div 5 = 10 \)
- \( 10 \div 5 = 2 \)

So, the prime factorization of 24000 is:
\[ 24000 = 2^6 \times 3^1 \times 5^3 \]

### Step 2: Analyze the Exponents
To determine if a number is a perfect cube, all the exponents in its prime factorization must be multiples of 3.

- For \( 2^6 \): The exponent is 6, which is a multiple of 3.
- For \( 3^1 \): The exponent is 1, which is not a multiple of 3.
- For \( 5^3 \): The exponent is 3, which is a multiple of 3.

### Step 3: Conclusion
Since not all exponents are multiples of 3 (specifically, the exponent of 3 is not), we conclude that 24000 is **not a perfect cube**.

### Summary of Solution
- Prime Factorization: \( 24000 = 2^6 \times 3^1 \times 5^3 \)
- Exponents: 6 (for 2), 1 (for 3), 3 (for 5)
- Conclusion: 24000 is not a perfect cube.

---