By what least number should 324 be multiplied to get a perfect cube?

Options

A. 12
B. 14
C. 16
D. 18

Answer

To determine the least number by which 324 should be multiplied to obtain a perfect cube, we can follow these steps:

### Step 1: Prime Factorization of 324
First, we need to find the prime factorization of 324.

- Divide 324 by 2:
\[ 324 \div 2 = 162 \]
- Divide 162 by 2:
\[ 162 \div 2 = 81 \]
- Divide 81 by 3:
\[ 81 \div 3 = 27 \]
- Divide 27 by 3:
\[ 27 \div 3 = 9 \]
- Divide 9 by 3:
\[ 9 \div 3 = 3 \]
- Divide 3 by 3:
\[ 3 \div 3 = 1 \]

Thus, the prime factorization of 324 is:
\[ 324 = 2^2 \times 3^4 \]

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

- The exponent of 2 is 2, which is not a multiple of 3. To make it a multiple of 3, we need one more 2 (i.e., \(2^1\)).
- The exponent of 3 is 4, which is also not a multiple of 3. To make it a multiple of 3, we need 2 more 3's (i.e., \(3^2\)).

### Step 3: Determine the Number to Multiply
To make 324 a perfect cube, we need to multiply it by:
\[ 2^1 \times 3^2 = 2 \times 9 = 18 \]

### Final Answer
Thus, the least number by which 324 should be multiplied to get a perfect cube is **18**.

---