When the number `3^(99)` is divided by 4, the remainder is

Options

A. 4
B. 3
C. 1
D. 2

Answer

To find the remainder when \( 3^{99} \) is divided by \( 4 \), we can use modular arithmetic.

### Step-by-Step Solution:

1. **Identify the Base and Modulus**: We need to find \( 3^{99} \mod 4 \).

2. **Calculate Powers of 3 Modulo 4**:
- First, we calculate \( 3^1 \mod 4 \):
\[ 3^1 = 3 \quad \Rightarrow \quad 3 \mod 4 = 3 \]
- Next, calculate \( 3^2 \mod 4 \):
\[ 3^2 = 9 \quad \Rightarrow \quad 9 \mod 4 = 1 \]
- Now calculate \( 3^3 \mod 4 \):
\[ 3^3 = 27 \quad \Rightarrow \quad 27 \mod 4 = 3 \]
- Finally, calculate \( 3^4 \mod 4 \):
\[ 3^4 = 81 \quad \Rightarrow \quad 81 \mod 4 = 1 \]

3. **Observe the Pattern**: From the calculations, we see a pattern:
- \( 3^1 \mod 4 = 3 \)
- \( 3^2 \mod 4 = 1 \)
- \( 3^3 \mod 4 = 3 \)
- \( 3^4 \mod 4 = 1 \)

The pattern alternates between \( 3 \) and \( 1 \):
- For odd powers of \( 3 \), the result is \( 3 \).
- For even powers of \( 3 \), the result is \( 1 \).

4. **Determine the Power**: Since \( 99 \) is odd, we can conclude:
\[ 3^{99} \mod 4 = 3 \]

5. **Final Answer**: The remainder when \( 3^{99} \) is divided by \( 4 \) is \( 3 \).

### Summary:
Thus, the remainder when \( 3^{99} \) is divided by \( 4 \) is \( 3 \).