Find out the unit digit in `(3259)^(1214)`

Options

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

Answer

To find the unit digit of \( (3259)^{1214} \), we can follow these steps:

### Step 1: Identify the unit digit of the base
The unit digit of \( 3259 \) is \( 9 \).

### Step 2: Analyze the pattern of unit digits for powers of \( 9 \)
Next, we need to find the unit digit of \( 9^n \) for various values of \( n \):
- \( 9^1 = 9 \) (unit digit is \( 9 \))
- \( 9^2 = 81 \) (unit digit is \( 1 \))
- \( 9^3 = 729 \) (unit digit is \( 9 \))
- \( 9^4 = 6561 \) (unit digit is \( 1 \))

From this, we can see that the unit digits of powers of \( 9 \) alternate:
- When \( n \) is odd, the unit digit is \( 9 \).
- When \( n \) is even, the unit digit is \( 1 \).

### Step 3: Determine whether the exponent is odd or even
The exponent given is \( 1214 \). Since \( 1214 \) is an even number, we can conclude that the unit digit of \( 9^{1214} \) will be \( 1 \).

### Step 4: Conclusion
Thus, the unit digit of \( (3259)^{1214} \) is \( 1 \).

### Final Answer
The unit digit is \( 1 \).

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