By what least number should 1536 be multiplied to get a perfect cube?

Options

A. 3
B. 4
C. 6
D. 8

Answer

To find the least number by which 1536 should be multiplied to get a perfect cube, we can follow these steps:

### Step 1: Prime Factorization of 1536
First, we need to find the prime factors of 1536. We can do this by continuously dividing by the smallest prime number (2) until we reach 1.

- 1536 ÷ 2 = 768
- 768 ÷ 2 = 384
- 384 ÷ 2 = 192
- 192 ÷ 2 = 96
- 96 ÷ 2 = 48
- 48 ÷ 2 = 24
- 24 ÷ 2 = 12
- 12 ÷ 2 = 6
- 6 ÷ 2 = 3
- 3 ÷ 3 = 1

So, the prime factorization of 1536 is:
\[ 1536 = 2^9 \times 3^1 \]

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

- For \(2^9\): The exponent 9 is already a multiple of 3.
- For \(3^1\): The exponent 1 is not a multiple of 3.

### Step 3: Determine the Required Multiplication
To make the exponent of 3 a multiple of 3, we need to increase it from 1 to the next multiple of 3, which is 3.

To do this, we need to multiply by \(3^{(3-1)} = 3^2 = 9\).

### Step 4: Conclusion
Thus, the least number by which 1536 should be multiplied to get a perfect cube is:
\[ \text{Least number} = 9 \]

### Final Answer
The least number that should be multiplied to 1536 to get a perfect cube is **9**.

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