`N = 55^3 + 17^3 - 72^3`, then N is divisible by :

Options

A. 3 & 17
B. 40 & 11
C. 11 & 15
D. all of these

Answer

To solve the problem \( N = 55^3 + 17^3 - 72^3 \) and determine its divisibility, we can use the identity for the sum of cubes.

### Step-by-Step Solution:

1. **Identify the Expression**:
We start with the expression:
\[ N = 55^3 + 17^3 - 72^3 \]

2. **Recognize the Sum of Cubes**:
We can use the identity for the sum of cubes:
\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]
In our case, let \( a = 55 \) and \( b = 17 \).

3. **Calculate \( a + b \)**:
First, we find:
\[ a + b = 55 + 17 = 72 \]

4. **Rewrite the Expression**:
Now we can rewrite \( N \) using the identity:
\[ N = (55 + 17)(55^2 - 55 \cdot 17 + 17^2) - 72^3 \]
Since \( 55 + 17 = 72 \), we have:
\[ N = 72(55^2 - 55 \cdot 17 + 17^2) - 72^3 \]

5. **Factor Out \( 72 \)**:
We can factor \( 72 \) out of the expression:
\[ N = 72 \left( (55^2 - 55 \cdot 17 + 17^2) - 72^2 \right) \]

6. **Simplify the Remaining Expression**:
Now, we need to simplify:
\[ 55^2 - 55 \cdot 17 + 17^2 - 72^2 \]
Calculate each term:
- \( 55^2 = 3025 \)
- \( 17^2 = 289 \)
- \( 72^2 = 5184 \)
- \( 55 \cdot 17 = 935 \)

Now substituting these values:
\[ 3025 - 935 + 289 - 5184 \]
Simplifying this gives:
\[ 3025 - 935 = 2090 \]
\[ 2090 + 289 = 2379 \]
\[ 2379 - 5184 = -2805 \]

7. **Final Expression for N**:
Thus, we have:
\[ N = 72 \cdot (-2805) \]

8. **Divisibility**:
Since \( N \) is a product of \( 72 \) and another integer, \( N \) is divisible by \( 72 \). Additionally, since \( 72 = 8 \times 9 \), it is also divisible by \( 8 \) and \( 9 \).

### Conclusion:
The final answer is that \( N \) is divisible by \( 72 \), \( 8 \), and \( 9 \).