If `N=4^(11)+4^(12)+4^(13)+4^(14)`, then how many positive factors of N are there ?

Options

A. 92
B. 48
C. 50
D. 51

Answer

To solve the problem, we need to simplify the expression for \( N \) and then find the number of positive factors of \( N \).

### Step 1: Rewrite \( N \)
We start with the expression:
\[ N = 4^{11} + 4^{12} + 4^{13} + 4^{14} \]
We can factor out the smallest power of \( 4 \), which is \( 4^{11} \):
\[ N = 4^{11}(1 + 4 + 4^2 + 4^3) \]

### Step 2: Simplify the expression in parentheses
Next, we simplify the expression inside the parentheses:
\[ 1 + 4 + 4^2 + 4^3 = 1 + 4 + 16 + 64 \]
Calculating this gives:
\[ 1 + 4 = 5 \]
\[ 5 + 16 = 21 \]
\[ 21 + 64 = 85 \]
So, we have:
\[ N = 4^{11} \cdot 85 \]

### Step 3: Factor \( 4^{11} \) and \( 85 \)
Now, we need to express \( N \) in terms of its prime factors:
\[ 4^{11} = (2^2)^{11} = 2^{22} \]
Next, we factor \( 85 \):
\[ 85 = 5 \times 17 \]
Thus, we can write \( N \) as:
\[ N = 2^{22} \cdot 5^1 \cdot 17^1 \]

### Step 4: Use the formula for the number of factors
To find the number of positive factors of \( N \), we use the formula:
If \( N = p_1^{e_1} \cdot p_2^{e_2} \cdots p_k^{e_k} \), then the number of positive factors \( d(N) \) is given by:
\[ d(N) = (e_1 + 1)(e_2 + 1) \cdots (e_k + 1) \]
In our case:
- \( e_1 = 22 \) (for \( 2^{22} \))
- \( e_2 = 1 \) (for \( 5^1 \))
- \( e_3 = 1 \) (for \( 17^1 \))

Calculating the number of factors:
\[ d(N) = (22 + 1)(1 + 1)(1 + 1) = 23 \cdot 2 \cdot 2 \]
\[ d(N) = 23 \cdot 4 = 92 \]

### Final Answer
Thus, the number of positive factors of \( N \) is \( \boxed{92} \).