`3^(41)xx7^(42)xx8^(43)` . find unit digit

Answer

To find the unit digit of the expression \(3^{41} \times 7^{42} \times 8^{43}\), we can break it down into steps.

### Step 1: Find the unit digit of \(3^{41}\)
The unit digits of powers of 3 follow a cyclic pattern:
- \(3^1 = 3\) (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)

This cycle (3, 9, 7, 1) repeats every 4 terms. To find the unit digit of \(3^{41}\), we calculate \(41 \mod 4\):
\[ 41 \div 4 = 10 \quad \text{remainder } 1 \]
Thus, \(41 \mod 4 = 1\). The unit digit of \(3^{41}\) corresponds to the first term in the cycle, which is 3.

### Step 2: Find the unit digit of \(7^{42}\)
The unit digits of powers of 7 also follow a cyclic pattern:
- \(7^1 = 7\) (unit digit is 7)
- \(7^2 = 49\) (unit digit is 9)
- \(7^3 = 343\) (unit digit is 3)
- \(7^4 = 2401\) (unit digit is 1)

This cycle (7, 9, 3, 1) repeats every 4 terms. To find the unit digit of \(7^{42}\), we calculate \(42 \mod 4\):
\[ 42 \div 4 = 10 \quad \text{remainder } 2 \]
Thus, \(42 \mod 4 = 2\). The unit digit of \(7^{42}\) corresponds to the second term in the cycle, which is 9.

### Step 3: Find the unit digit of \(8^{43}\)
The unit digits of powers of 8 follow a cyclic pattern:
- \(8^1 = 8\) (unit digit is 8)
- \(8^2 = 64\) (unit digit is 4)
- \(8^3 = 512\) (unit digit is 2)
- \(8^4 = 4096\) (unit digit is 6)

This cycle (8, 4, 2, 6) repeats every 4 terms. To find the unit digit of \(8^{43}\), we calculate \(43 \mod 4\):
\[ 43 \div 4 = 10 \quad \text{remainder } 3 \]
Thus, \(43 \mod 4 = 3\). The unit digit of \(8^{43}\) corresponds to the third term in the cycle, which is 2.

### Step 4: Combine the unit digits
Now we have the unit digits:
- Unit digit of \(3^{41}\) is 3.
- Unit digit of \(7^{42}\) is 9.
- Unit digit of \(8^{43}\) is 2.

Now we multiply these unit digits together:
\[ 3 \times 9 = 27 \quad \text{(unit digit is 7)} \]
Then multiply the result by the unit digit of \(8^{43}\):
\[ 7 \times 2 = 14 \quad \text{(unit digit is 4)} \]

### Final Answer
The unit digit of \(3^{41} \times 7^{42} \times 8^{43}\) is **4**.

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