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

Answer

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

### Step 1: Rewrite \( 2^{403} \)
We can express \( 2^{403} \) in a more manageable form:
\[ 2^{403} = 2^{400} \cdot 2^3 \]

### Step 2: Simplify \( 2^{400} \)
Next, we can rewrite \( 2^{400} \):
\[ 2^{400} = (2^4)^{100} = 16^{100} \]

### Step 3: Substitute back into the fraction
Now we substitute back into our fraction:
\[ \frac{2^{403}}{15} = \frac{16^{100} \cdot 2^3}{15} = \frac{16^{100} \cdot 8}{15} \]

### Step 4: Analyze \( 16^{100} \)
We can express \( 16^{100} \) in terms of its modulo with respect to 15:
\[ 16 \equiv 1 \mod 15 \]
Thus,
\[ 16^{100} \equiv 1^{100} \equiv 1 \mod 15 \]

### Step 5: Substitute back into the fraction
Now, substituting this back, we have:
\[ \frac{16^{100} \cdot 8}{15} \equiv \frac{1 \cdot 8}{15} \equiv \frac{8}{15} \mod 15 \]

### Step 6: Identify the fractional part
The fractional part of \( \frac{2^{403}}{15} \) is thus:
\[ \frac{8}{15} \]

### Step 7: Relate to \( k \)
From the problem statement, we know that the fractional part can be expressed as \( \frac{k}{15} \):
\[ \frac{k}{15} = \frac{8}{15} \]
This implies that:
\[ k = 8 \]

### Conclusion
Therefore, the value of \( k \) is:
\[ \boxed{8} \]