The digit in unit's place of the product `(2153)^(167)` is :

Options

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

Answer

To find the digit in the unit's place of the product \( (2153)^{167} \), we can follow these steps:

### Step 1: Identify the unit digit of the base
The base number is \( 2153 \). The unit digit of \( 2153 \) is \( 3 \).

### Step 2: Focus on the unit digit
We need to find the unit digit of \( 3^{167} \) since the unit digit of \( (2153)^{167} \) will be the same as the unit digit of \( 3^{167} \).

### Step 3: Determine the pattern of unit digits for powers of 3
Let's calculate the unit digits of the first few powers of \( 3 \):
- \( 3^1 = 3 \) (unit digit is \( 3 \))
- \( 3^2 = 9 \) (unit digit is \( 9 \))
- \( 3^3 = 27 \) (unit digit is \( 7 \))
- \( 3^4 = 81 \) (unit digit is \( 1 \))

Now, we can see that the unit digits repeat every 4 powers:
- \( 3, 9, 7, 1 \)

### Step 4: Find the relevant power
To find the unit digit of \( 3^{167} \), we need to determine the position of \( 167 \) in the cycle of \( 4 \). We do this by calculating \( 167 \mod 4 \):
\[ 167 \div 4 = 41 \quad \text{(remainder } 3\text{)} \]
Thus, \( 167 \mod 4 = 3 \).

### Step 5: Identify the unit digit based on the remainder
From our pattern:
- If the remainder is \( 1 \), the unit digit is \( 3 \).
- If the remainder is \( 2 \), the unit digit is \( 9 \).
- If the remainder is \( 3 \), the unit digit is \( 7 \).
- If the remainder is \( 0 \), the unit digit is \( 1 \).

Since the remainder is \( 3 \), the unit digit of \( 3^{167} \) is \( 7 \).

### Final Answer
Therefore, the unit digit of \( (2153)^{167} \) is \( 7 \).

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