Find the smallest number by which 1323 must be multiplied so that the product is a perfect cube.

Options

A. 5
B. 6
C. 7
D. 8

Answer

To find the smallest number by which 1323 must be multiplied to make it a perfect cube, we can follow these steps:

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

- Divide 1323 by 3 (the smallest prime number):
\[ 1323 \div 3 = 441 \]
- Divide 441 by 3:
\[ 441 \div 3 = 147 \]
- Divide 147 by 3:
\[ 147 \div 3 = 49 \]
- Divide 49 by 7:
\[ 49 \div 7 = 7 \]
- Divide 7 by 7:
\[ 7 \div 7 = 1 \]

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

### Step 2: Identify the Exponents
In the factorization \(3^3 \times 7^2\), the exponents of the prime factors are:
- For 3: 3 (which is already a perfect cube)
- For 7: 2 (which is not a perfect cube)

### Step 3: Make the Exponents Multiples of 3
To make the number a perfect cube, all the exponents in the prime factorization must be multiples of 3.

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

### Step 4: Determine the Number to Multiply
To increase the exponent of 7 from 2 to 3, we need one more factor of 7. Therefore, we need to multiply 1323 by \(7^1\).

### Conclusion
Thus, the smallest number by which 1323 must be multiplied to make it a perfect cube is:
\[ \boxed{7} \]

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