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To find the unit digit of \((137^{13})^{47}\), we can simplify this problem by focusing only on the unit digits. Here’s a step-by-step solution:

1. **Identify the Unit Digit of the Base**: 
   The unit digit of \(137\) is \(7\). So, we only need to consider the powers of \(7\).

2. **Determine the Pattern in Powers of \(7\)**:
   The unit digits of powers of \(7\) follow a repeating cycle. Let's examine the first few powers:
   
   - \(7^1 = 7\), so the unit digit is \(7\).
   - \(7^2 = 49\), so the unit digit is \(9\).
   - \(7^3 = 343\), so the unit digit is \(3\).
   - \(7^4 = 2401\), so the unit digit is \(1\).

   Thus, the unit digits of powers of \(7\) cycle every 4 terms: \(7, 9, 3, 1\).

3. **Reduce the Power Modulo 4**:
   We now need to find the unit digit of \(7^{13 \times 47}\). Simplify \(13 \times 47\) modulo 4, since the cycle length is 4.

   \[
   13 \times 47 = 611
   \]

   Now, find \(611 \mod 4\):
   \[
   611 \div 4 = 152\text{ remainder }3
   \]
   Therefore, \(611 \equiv 3 \mod 4\).

4. **Determine the Unit Digit**:
   Since \(611 \equiv 3 \mod 4\), the unit digit of \(7^{611}\) will be the same as the unit digit of \(7^3\), which is \(3\).

**Answer**: The unit digit of \((137^{13})^{47}\) is **3**.

Would you like more details or have any questions?

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Here are some related questions for practice:

1. What is the unit digit of \(27^{53}\)?
2. How would the unit digit change if the exponent was even?
3. Can you explain why cycles occur in powers of integers?
4. What if the base number was \(1377\) instead of \(137\)?
5. How do you find unit digits of expressions like \((125^5)^{10}\)?

**Tip**: Finding unit digits often involves identifying cycles in the powers of single-digit bases.

Learn how to find the unit digit of the expression (137^13)^47 by identifying cyclic patterns in powers of integers. This solution uses modular arithmetic and exponentiation to simplify calculations. Ideal for high school students studying advanced number theory.

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