What is the greatest prime factor of `(2^4)^(2) - 1` ?

Options

A. 3
B. 5
C. 11
D. 17

Answer

To find the greatest prime factor of \( (2^4)^2 - 1 \), we can follow these steps:

### Step 1: Simplify the expression
We start with the expression \( (2^4)^2 - 1 \).

Using the exponent rule \( (a^m)^n = a^{m \cdot n} \):
\[ (2^4)^2 = 2^{4 \cdot 2} = 2^8 \]
So, we rewrite the expression as:
\[ 2^8 - 1 \]

### Step 2: Calculate \( 2^8 \)
Now, we calculate \( 2^8 \):
\[ 2^8 = 256 \]
Thus, we have:
\[ 2^8 - 1 = 256 - 1 = 255 \]

### Step 3: Factor 255
Next, we need to find the prime factors of 255. We can start by dividing 255 by the smallest prime numbers.

1. **Dividing by 3**:
\[ 255 \div 3 = 85 \]
So, we have \( 255 = 3 \times 85 \).

2. **Factoring 85**:
Now we factor 85. We can divide 85 by 5:
\[ 85 \div 5 = 17 \]
Thus, we have \( 85 = 5 \times 17 \).

Putting it all together, we get:
\[ 255 = 3 \times 5 \times 17 \]

### Step 4: Identify the greatest prime factor
The prime factors of 255 are 3, 5, and 17. Among these, the greatest prime factor is:
\[ \text{Greatest prime factor} = 17 \]

### Final Answer
Therefore, the greatest prime factor of \( (2^4)^2 - 1 \) is:
\[ \boxed{17} \]

---