Find the smallest number by which 392 must be multiplied so that the product is a perfect cube.

Options

A. 5
B. 3
C. 2
D. 7

Answer

To find the smallest number by which 392 must be multiplied so that the product is a perfect cube, we can follow these steps:

### Step 1: Prime Factorization of 392
First, we need to perform the prime factorization of 392.

- Divide 392 by 2:
\[ 392 \div 2 = 196 \]
- Divide 196 by 2:
\[ 196 \div 2 = 98 \]
- Divide 98 by 2:
\[ 98 \div 2 = 49 \]
- Divide 49 by 7:
\[ 49 \div 7 = 7 \]
- Divide 7 by 7:
\[ 7 \div 7 = 1 \]

So, the prime factorization of 392 is:
\[ 392 = 2^3 \times 7^2 \]

### Step 2: Analyze the Exponents
Next, we need to analyze the exponents of the prime factors to determine how many of each factor we need to make the product a perfect cube.

- For \(2^3\): The exponent is 3, which is already a multiple of 3 (since \(3 \div 3 = 1\)).
- For \(7^2\): The exponent is 2, which is not a multiple of 3. To make it a multiple of 3, we need to increase it to 3.

### Step 3: Determine the Required Factor
To make \(7^2\) into a perfect cube, we need to increase the exponent from 2 to 3. This means we need one more factor of 7.

Thus, we need to multiply 392 by \(7^1\) to make it a perfect cube.

### Step 4: Final Calculation
The smallest number by which 392 must be multiplied is:
\[ 7 \]

### Conclusion
Therefore, the answer is:
\[ \text{The smallest number is } 7. \]

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