What is the smallest number by which `3087` may be multiplied so that the product is a perfect cube?

Options

A. `2`
B. `3`
C. `1`
D. `7`

Answer

To find the smallest number by which 3087 must be multiplied to make it a perfect cube, we will follow these steps:

### Step 1: Prime Factorization of 3087
First, we need to find the prime factorization of 3087.

- Divide 3087 by 3:
\[ 3087 \div 3 = 1029 \]
- Divide 1029 by 3:
\[ 1029 \div 3 = 343 \]
- Divide 343 by 7:
\[ 343 \div 7 = 49 \]
- Divide 49 by 7:
\[ 49 \div 7 = 7 \]
- Finally, divide 7 by 7:
\[ 7 \div 7 = 1 \]

So, the prime factorization of 3087 is:
\[ 3087 = 3^2 \times 7^3 \]

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

### Step 3: Determine the Required Multiplication
For a number to be a perfect cube, all the exponents in its prime factorization must be multiples of \(3\).

- The exponent of \(3\) is \(2\). To make it a multiple of \(3\), we need to increase it to \(3\). Therefore, we need to multiply by \(3^{(3-2)} = 3^1 = 3\).
- The exponent of \(7\) is already \(3\), which is a multiple of \(3\), so we do not need to multiply by \(7\).

### Step 4: Calculate the Smallest Number
Thus, the smallest number by which \(3087\) must be multiplied to make it a perfect cube is:
\[ 3 \]

### Final Answer
The smallest number by which \(3087\) may be multiplied so that the product is a perfect cube is \(3\).

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