Find the unit digit in the expression : (3^57*6^41*7^63)

Options

A. 3
B. 4
C. 6
D. 9

Answer

To find the unit digit of the expression \(3^{57} \times 6^{41} \times 7^{63}\), we can break it down into three parts and find the unit digit of each part separately.

### Step 1: Find the unit digit of \(3^{57}\)

The unit digits of the powers of 3 follow a pattern:
- \(3^1 = 3\) (unit digit 3)
- \(3^2 = 9\) (unit digit 9)
- \(3^3 = 27\) (unit digit 7)
- \(3^4 = 81\) (unit digit 1)

This pattern repeats every 4 terms: 3, 9, 7, 1.

To find the unit digit of \(3^{57}\), we calculate \(57 \mod 4\):
\[ 57 \div 4 = 14 \quad \text{remainder } 1 \]
So, \(57 \mod 4 = 1\). Therefore, the unit digit of \(3^{57}\) is the same as the unit digit of \(3^1\), which is **3**.

### Step 2: Find the unit digit of \(6^{41}\)

The unit digit of any power of 6 is always 6:
- \(6^1 = 6\)
- \(6^2 = 36\) (unit digit 6)
- \(6^3 = 216\) (unit digit 6)
- ...

Thus, the unit digit of \(6^{41}\) is **6**.

### Step 3: Find the unit digit of \(7^{63}\)

The unit digits of the powers of 7 also follow a pattern:
- \(7^1 = 7\) (unit digit 7)
- \(7^2 = 49\) (unit digit 9)
- \(7^3 = 343\) (unit digit 3)
- \(7^4 = 2401\) (unit digit 1)

This pattern repeats every 4 terms: 7, 9, 3, 1.

To find the unit digit of \(7^{63}\), we calculate \(63 \mod 4\):
\[ 63 \div 4 = 15 \quad \text{remainder } 3 \]
So, \(63 \mod 4 = 3\). Therefore, the unit digit of \(7^{63}\) is the same as the unit digit of \(7^3\), which is **3**.

### Step 4: Combine the unit digits

Now we can combine the unit digits we found:
- Unit digit of \(3^{57}\) is **3**
- Unit digit of \(6^{41}\) is **6**
- Unit digit of \(7^{63}\) is **3**

Now we multiply these unit digits together:
\[ 3 \times 6 \times 3 \]

Calculating step by step:
1. \(3 \times 6 = 18\) (unit digit 8)
2. \(8 \times 3 = 24\) (unit digit 4)

Thus, the unit digit of the entire expression \(3^{57} \times 6^{41} \times 7^{63}\) is **4**.

### Final Answer
The unit digit of the expression \(3^{57} \times 6^{41} \times 7^{63}\) is **4**.

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