The square of a natural number subtracted from its cube is 48. The number is :

Options

A. 8
B. 6
C. 5
D. 4

Answer

To solve the problem step by step, we need to find a natural number \( n \) such that the cube of \( n \) minus the square of \( n \) equals 48.

### Step 1: Set up the equation
We can express the problem mathematically as:
\[ n^3 - n^2 = 48 \]

### Step 2: Rearrange the equation
Rearranging the equation gives us:
\[ n^3 - n^2 - 48 = 0 \]

### Step 3: Factor out \( n^2 \)
We can factor out \( n^2 \) from the left side:
\[ n^2(n - 1) = 48 \]

### Step 4: Test natural numbers
Since we are looking for natural numbers, we can test small values of \( n \) to see if they satisfy the equation.

1. **For \( n = 1 \)**:
\[ 1^3 - 1^2 = 1 - 1 = 0 \quad (\text{Not } 48) \]

2. **For \( n = 2 \)**:
\[ 2^3 - 2^2 = 8 - 4 = 4 \quad (\text{Not } 48) \]

3. **For \( n = 3 \)**:
\[ 3^3 - 3^2 = 27 - 9 = 18 \quad (\text{Not } 48) \]

4. **For \( n = 4 \)**:
\[ 4^3 - 4^2 = 64 - 16 = 48 \quad (\text{This works!}) \]

### Step 5: Conclusion
The natural number we are looking for is:
\[ \boxed{4} \]