What is the minimum number you should divide 23958 by to get a perfect cube?

Options

A. 6
B. 12
C. 18
D. 24

Answer

To find the minimum number you should divide 23958 by to get a perfect cube, we will follow these steps:

### Step 1: Prime Factorization of 23958
We will start by finding the prime factorization of 23958.

1. **Divide by 2** (the smallest prime number):
- 23958 ÷ 2 = 11979
- So, we have one factor of 2.

2. **Divide 11979 by 3** (the next smallest prime number):
- 11979 ÷ 3 = 3993
- 3993 ÷ 3 = 1331
- So, we have two factors of 3.

3. **Now, factor 1331**:
- 1331 = 11 × 11 × 11 = 11^3
- So, we have three factors of 11.

Thus, the complete prime factorization of 23958 is:
\[ 23958 = 2^1 \times 3^2 \times 11^3 \]

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

- For \(2^1\): The exponent is 1, which is not a multiple of 3. We need to increase it to 3. Thus, we need to divide by \(2^{1}\) (to remove one factor of 2).
- For \(3^2\): The exponent is 2, which is also not a multiple of 3. We need to increase it to 3. Thus, we need to divide by \(3^{2}\) (to remove two factors of 3).
- For \(11^3\): The exponent is already a multiple of 3.

### Step 3: Calculate the Minimum Number to Divide
To make the exponents multiples of 3, we need to divide by:
\[ 2^1 \times 3^2 = 2 \times 9 = 18 \]

### Final Answer
The minimum number you should divide 23958 by to get a perfect cube is **18**.

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