`8^6 - 5^6` is individually divisible by :

Options

A. 91
B. 49
C. 129
D. all of these

Answer

To solve the expression \(8^6 - 5^6\) and determine its divisibility, we can use the difference of cubes formula. Here’s a step-by-step breakdown:

### Step 1: Recognize the expression as a difference of cubes
We can rewrite \(8^6 - 5^6\) as \((8^2)^3 - (5^2)^3\). This gives us:
\[ 8^6 - 5^6 = (64)^3 - (25)^3 \]

### Step 2: Apply the difference of cubes formula
The difference of cubes can be factored using the formula:
\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]
In our case, let \(a = 64\) and \(b = 25\):
\[ (64 - 25)(64^2 + 64 \cdot 25 + 25^2) \]

### Step 3: Calculate \(64 - 25\)
Calculating \(64 - 25\):
\[ 64 - 25 = 39 \]

### Step 4: Calculate \(64^2 + 64 \cdot 25 + 25^2\)
Now we calculate \(64^2\), \(64 \cdot 25\), and \(25^2\):
\[ 64^2 = 4096 \]
\[ 64 \cdot 25 = 1600 \]
\[ 25^2 = 625 \]
Now, add these values together:
\[ 4096 + 1600 + 625 = 6321 \]

### Step 5: Combine the results
Now, we can express \(8^6 - 5^6\) as:
\[ 8^6 - 5^6 = 39 \cdot 6321 \]

### Step 6: Determine the divisibility
Now we need to check the divisibility of both factors:
1. **For 39**: The prime factorization of 39 is \(3 \times 13\).
2. **For 6321**: We can check if it is divisible by smaller prime numbers.

### Step 7: Check divisibility of 6321
To check if 6321 is divisible by 3:
- Sum of digits: \(6 + 3 + 2 + 1 = 12\) (which is divisible by 3).
Thus, 6321 is divisible by 3.

To check if 6321 is divisible by 13:
- Performing the division \(6321 \div 13 = 487\) (which is an integer).
Thus, 6321 is divisible by 13.

### Conclusion
Both \(39\) and \(6321\) are divisible by \(3\) and \(13\). Therefore, \(8^6 - 5^6\) is divisible by \(3\) and \(13\).