By what least number should 648 be multiplied to get a perfect cube?

Options

A. 3
B. 6
C. 9
D. 8

Answer

To determine the least number by which 648 should be multiplied to obtain a perfect cube, we can follow these steps:

### Step 1: Prime Factorization of 648
First, we need to find the prime factorization of 648.

648 can be divided by 2:
- 648 ÷ 2 = 324
- 324 ÷ 2 = 162
- 162 ÷ 2 = 81
- 81 ÷ 3 = 27
- 27 ÷ 3 = 9
- 9 ÷ 3 = 3
- 3 ÷ 3 = 1

So, the prime factorization of 648 is:
\[ 648 = 2^3 \times 3^4 \]

### Step 2: Grouping the Prime Factors
To form a perfect cube, each prime factor's exponent must be a multiple of 3.

- For \(2^3\), the exponent is already 3 (which is a multiple of 3).
- For \(3^4\), the exponent is 4. To make this a multiple of 3, we need to increase it to 6 (the next multiple of 3).

### Step 3: Determine the Additional Factors Needed
To increase the exponent of \(3^4\) to \(3^6\), we need to multiply by \(3^{(6-4)} = 3^2 = 9\).

### Step 4: Conclusion
Thus, the least number by which 648 should be multiplied to get a perfect cube is **9**.

### Final Answer
The least number to be multiplied is **9**.

---