What is the unit digit of` (12343)^ 12341`?

Options

A. 9
B. 3
C. 7
D. 1

Answer

To find the unit digit of \( (12343)^{12341} \), we can follow these steps:

### Step 1: Identify the unit digit of the base
The first step is to find the unit digit of the base number, which is \( 12343 \). The unit digit is simply the last digit of the number.

**Unit digit of 12343 = 3**

### Step 2: Determine the power
Next, we need to consider the exponent, which is \( 12341 \).

### Step 3: Find the cycle of unit digits for powers of 3
Now, we need to find the unit digits of the powers of 3, as the unit digit of \( (12343)^{12341} \) will be the same as the unit digit of \( 3^{12341} \). The unit digits of powers of 3 form a repeating cycle:

- \( 3^1 = 3 \) (unit digit = 3)
- \( 3^2 = 9 \) (unit digit = 9)
- \( 3^3 = 27 \) (unit digit = 7)
- \( 3^4 = 81 \) (unit digit = 1)
- \( 3^5 = 243 \) (unit digit = 3) - cycle repeats

The cycle of unit digits for powers of 3 is: **3, 9, 7, 1** and it repeats every 4 terms.

### Step 4: Find the position in the cycle
To find the correct unit digit for \( 3^{12341} \), we need to find the position of \( 12341 \) in the cycle. We do this by calculating \( 12341 \mod 4 \):

\[ 12341 \div 4 = 3085 \quad \text{(remainder 1)} \]

Thus, \( 12341 \mod 4 = 1 \).

### Step 5: Determine the unit digit based on the cycle
Since the remainder is 1, we look at the first position in our cycle of unit digits:

- Position 1 corresponds to the unit digit of \( 3^1 \), which is **3**.

### Conclusion
Therefore, the unit digit of \( (12343)^{12341} \) is **3**.

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