What is the smallest natural number by which 7056 must be divided so that the quotient becomes a perfect cube?

Options

A. 882
B. 441
C. 294
D. 126

Answer

To find the smallest natural number by which 7056 must be divided so that the quotient becomes a perfect cube, we can follow these steps:

### Step 1: Prime Factorization of 7056
We start by finding the prime factors of 7056.

1. Divide 7056 by 2:
\[ 7056 \div 2 = 3528 \]
2. Divide 3528 by 2:
\[ 3528 \div 2 = 1764 \]
3. Divide 1764 by 2:
\[ 1764 \div 2 = 882 \]
4. Divide 882 by 2:
\[ 882 \div 2 = 441 \]
5. Divide 441 by 3:
\[ 441 \div 3 = 147 \]
6. Divide 147 by 3:
\[ 147 \div 3 = 49 \]
7. Divide 49 by 7:
\[ 49 \div 7 = 7 \]
8. Finally, divide 7 by 7:
\[ 7 \div 7 = 1 \]

So, the prime factorization of 7056 is:
\[ 7056 = 2^4 \times 3^2 \times 7^2 \]

### Step 2: Determine the Exponents for Perfect Cube
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 4 (not a multiple of 3).
- The exponent of 3 is 2 (not a multiple of 3).
- The exponent of 7 is 2 (not a multiple of 3).

### Step 3: Adjust the Exponents
To make each exponent a multiple of 3, we need to adjust them:

- For \(2^4\), we need to reduce it to \(2^3\). This means we need to remove \(1\) factor of \(2\).
- For \(3^2\), we need to increase it to \(3^3\). This means we need to add \(1\) factor of \(3\).
- For \(7^2\), we need to increase it to \(7^3\). This means we need to add \(1\) factor of \(7\).

### Step 4: Calculate the Number to Divide
The smallest natural number we need to divide by is given by the product of the factors we need to remove or add:

- Remove \(2^1\) → \(2\)
- Add \(3^1\) → \(3\)
- Add \(7^1\) → \(7\)

Thus, the number we need to divide by is:
\[ 2 \times 3 \times 7 = 42 \]

### Step 5: Final Calculation
Now, we need to check if dividing 7056 by 42 gives us a perfect cube:

\[ 7056 \div 42 = 168 \]

Now, we can factor 168 to check if it is a perfect cube:
\[ 168 = 2^3 \times 3^1 \times 7^1 \]

This is not a perfect cube since the exponents are not multiples of 3.

### Conclusion
To find the smallest natural number by which 7056 must be divided to yield a perfect cube, we need to divide by \(882\) (which is \(2^3 \times 3^2 \times 7^2\)).

Thus, the required answer is:
\[ \text{The smallest natural number is } 882. \]