Find the unit digit of `3^(47) + 7^(52)`.

Answer

To find the unit digit of \(3^{47} + 7^{52}\), we can follow these steps:

### Step 1: Find the unit digit of \(3^{47}\)
The unit digits of powers of 3 follow a 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)
- \(3^5 = 243\) (unit digit is 3)

The unit digits repeat every 4 terms: 3, 9, 7, 1.

To find the unit digit of \(3^{47}\), we calculate \(47 \mod 4\):
\[ 47 \div 4 = 11 \quad \text{remainder } 3 \]
So, \(47 \mod 4 = 3\).

This means the unit digit of \(3^{47}\) corresponds to the unit digit of \(3^3\), which is 7.

### Step 2: Find the unit digit of \(7^{52}\)
The unit digits of powers of 7 also follow a 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)
- \(7^5 = 16807\) (unit digit is 7)

The unit digits repeat every 4 terms: 7, 9, 3, 1.

To find the unit digit of \(7^{52}\), we calculate \(52 \mod 4\):
\[ 52 \div 4 = 13 \quad \text{remainder } 0 \]
So, \(52 \mod 4 = 0\).

This means the unit digit of \(7^{52}\) corresponds to the unit digit of \(7^4\), which is 1.

### Step 3: Add the unit digits
Now we add the unit digits we found:
\[ \text{Unit digit of } 3^{47} = 7 \]
\[ \text{Unit digit of } 7^{52} = 1 \]
\[ \text{Sum} = 7 + 1 = 8 \]

### Conclusion
The unit digit of \(3^{47} + 7^{52}\) is **8**.

---