Fractional part of the number `(3^(102))/(40)` is :

Options

A. `1/40`
B. `3/40`
C. `9/40`
D. `1/10`

Explanation

To find the fractional part of the number \( \frac{3^{102}}{40} \), we can follow these steps:

### Step 1: Find \( 3^{102} \mod 40 \)

First, we need to determine \( 3^{102} \mod 40 \). We can use the property of modular arithmetic to simplify our calculations.

### Step 2: Identify the pattern in powers of 3 modulo 40

Let's calculate the first few powers of 3 modulo 40:

- \( 3^1 = 3 \)
- \( 3^2 = 9 \)
- \( 3^3 = 27 \)
- \( 3^4 = 81 \equiv 1 \mod 40 \)

Notice that \( 3^4 \equiv 1 \mod 40 \). This means that every fourth power of 3 will cycle back to 1.

### Step 3: Reduce the exponent modulo 4

Since \( 3^4 \equiv 1 \mod 40 \), we can reduce the exponent \( 102 \) modulo \( 4 \):

\[ 102 \mod 4 = 2 \]

This means:

\[ 3^{102} \equiv 3^2 \mod 40 \]

### Step 4: Calculate \( 3^2 \mod 40 \)

Now we can calculate \( 3^2 \):

\[ 3^2 = 9 \]

Thus,

\[ 3^{102} \equiv 9 \mod 40 \]

### Step 5: Find the fractional part

Now we can express \( \frac{3^{102}}{40} \) as:

\[ \frac{3^{102}}{40} = \frac{40k + 9}{40} \text{ for some integer } k \]

The fractional part is given by:

\[ \text{Fractional part} = \frac{9}{40} \]

### Final Answer

Thus, the fractional part of \( \frac{3^{102}}{40} \) is:

\[ \frac{9}{40} \]

---