Find the remainder when `4^(13)` divided by 3?
A. 1
B. 2
C. 0
D. 3

Options

A. B
B. D
C. C
D. A

Answer

To find the remainder when \( 4^{13} \) is divided by 3, we can follow these steps:

### Step 1: Simplify \( 4 \) modulo \( 3 \)
First, we need to find the remainder when \( 4 \) is divided by \( 3 \):
\[ 4 \div 3 = 1 \quad \text{(remainder 1)} \]
So, \( 4 \equiv 1 \mod 3 \).

### Step 2: Substitute into the exponent
Now, we can substitute \( 4 \) with \( 1 \) in the expression \( 4^{13} \):
\[ 4^{13} \equiv 1^{13} \mod 3 \]

### Step 3: Calculate \( 1^{13} \)
Next, we calculate \( 1^{13} \):
\[ 1^{13} = 1 \]

### Step 4: Find the remainder
Thus, we find that:
\[ 4^{13} \equiv 1 \mod 3 \]
This means that when \( 4^{13} \) is divided by \( 3 \), the remainder is \( 1 \).

### Conclusion
The remainder when \( 4^{13} \) is divided by \( 3 \) is \( 1 \).

### Answer
The correct option is **A. 1**.

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