By what least number 1323 be multiplied to obtain a number which is perfect cube?

Options

A. a)2
B. b)3
C. c)5
D. d)7

Answer

To determine the least number by which 1323 must be multiplied to obtain a perfect cube, we will follow these steps:

### Step 1: Prime Factorization of 1323
First, we need to find the prime factorization of 1323.

1. **Divide by 3**:
- 1323 ÷ 3 = 441
2. **Divide by 3 again**:
- 441 ÷ 3 = 147
3. **Divide by 3 again**:
- 147 ÷ 3 = 49
4. **Divide by 7**:
- 49 ÷ 7 = 7
5. **Divide by 7 again**:
- 7 ÷ 7 = 1

So, the prime factorization of 1323 is:
\[ 1323 = 3^3 \times 7^2 \]

### Step 2: Identify the Exponents
Next, we look at the exponents of the prime factors:
- For \(3\), the exponent is \(3\) (which is already a perfect cube).
- For \(7\), the exponent is \(2\).

### Step 3: Make Exponents Multiples of 3
To form a perfect cube, all prime factors must have exponents that are multiples of 3.

- The exponent of \(3\) is already \(3\), which is a multiple of \(3\).
- The exponent of \(7\) is \(2\). To make \(2\) a multiple of \(3\), we need to increase it to \(3\).

### Step 4: Calculate the Required Multiplication
To change the exponent of \(7\) from \(2\) to \(3\), we need to multiply by \(7^{3-2} = 7^1 = 7\).

### Conclusion
Thus, the least number by which 1323 must be multiplied to obtain a perfect cube is **7**.