If the fractional part of the number `(2^(403))/(15)` is `(k)/(15)`, then `k` is equal to

Options

A. 14
B. 6
C. 4
D. 8

Explanation

To find the value of \( k \) such that the fractional part of \( \frac{2^{403}}{15} \) is \( \frac{k}{15} \), we can follow these steps:

### Step 1: Rewrite \( 2^{403} \)
We can express \( 2^{403} \) as:
\[ 2^{403} = 2^{400} \times 2^3 = (2^{100})^4 \times 8 \]

### Step 2: Simplify \( 2^{400} \)
We can simplify \( 2^{400} \) further:
\[ 2^{400} = (2^4)^{100} = 16^{100} \]
Thus, we have:
\[ 2^{403} = 8 \times 16^{100} \]

### Step 3: Express \( 16^{100} \) in terms of \( 15 \)
Next, we can express \( 16^{100} \) in a form that can be expanded using the binomial theorem:
\[ 16^{100} = (15 + 1)^{100} \]
Using the binomial theorem, we can expand this:
\[ (15 + 1)^{100} = \sum_{k=0}^{100} \binom{100}{k} 15^k \cdot 1^{100-k} \]

### Step 4: Identify the terms
The expansion gives us:
\[ (15 + 1)^{100} = 1 + \binom{100}{1} \cdot 15 + \binom{100}{2} \cdot 15^2 + \ldots + 15^{100} \]
The first term \( 1 \) is the integer part, and all other terms involve powers of \( 15 \).

### Step 5: Multiply by \( 8 \)
Now, multiplying the entire expansion by \( 8 \):
\[ 8 \cdot (15 + 1)^{100} = 8 + 8 \cdot \left( \binom{100}{1} \cdot 15 + \binom{100}{2} \cdot 15^2 + \ldots + 15^{100} \right) \]

### Step 6: Divide by \( 15 \)
Now, we divide the entire expression by \( 15 \):
\[ \frac{8 \cdot (15 + 1)^{100}}{15} = \frac{8}{15} + \frac{8 \cdot \left( \binom{100}{1} \cdot 15 + \binom{100}{2} \cdot 15^2 + \ldots + 15^{100} \right)}{15} \]
The first term \( \frac{8}{15} \) is the fractional part we are interested in.

### Step 7: Identify \( k \)
From the expression, we can see that the fractional part of \( \frac{2^{403}}{15} \) is \( \frac{8}{15} \). Therefore, \( k \) is equal to \( 8 \).

### Final Answer
Thus, the value of \( k \) is:
\[ \boxed{8} \]