What is the unit digit of the product (217)^413 × (819)^547 × (414)^624 × (342)^812?

Introduction / Context:
This question focuses on finding the unit digit of a very large product involving powers of multi digit numbers. Direct computation is impossible, so we rely on patterns in unit digits of powers and modular arithmetic concepts.

Given Data / Assumptions:
- We need the unit digit of (217)^413 × (819)^547 × (414)^624 × (342)^812.
- Only the unit digits of base numbers affect the unit digit of the product.
- Exponents are positive integers.

Concept / Approach:
The key idea is to reduce each base number to its unit digit and then analyze patterns in powers of that last digit. We then compute the unit digit of each powered term and multiply those unit digits, again focusing only on the unit digit of the product. Cycles for unit digits of powers of 2, 4, 7, and 9 are well known.

Step-by-Step Solution:
Step 1: Identify unit digits of bases: 217 has 7, 819 has 9, 414 has 4, and 342 has 2.Step 2: Consider 7^n. The unit digit cycle for 7 is 7, 9, 3, 1 with period 4. Since 413 mod 4 = 1, the unit digit of 217^413 is 7.Step 3: Consider 9^n. Powers of 9 alternate between 9 (odd exponents) and 1 (even exponents). Since 547 is odd, the unit digit of 819^547 is 9.Step 4: Consider 4^n. The unit digit cycle for 4 is 4, 6 with period 2. Since 624 is even, the unit digit of 414^624 is 6.Step 5: Consider 2^n. The unit digit cycle for 2 is 2, 4, 8, 6 with period 4. Since 812 mod 4 = 0, the unit digit of 342^812 is 6.Step 6: Now multiply the unit digits: 7 × 9 × 6 × 6.Step 7: First 7 × 9 has unit digit 3. Then 3 × 6 has unit digit 8. Finally 8 × 6 has unit digit 8.Step 8: Therefore the unit digit of the entire product is 8.

Verification / Alternative check:
We can verify the last steps of multiplication separately. 7 × 9 = 63 so unit digit 3. Then 3 × 6 = 18 so unit digit 8. Then 8 × 6 = 48 so unit digit 8. Since unit digit patterns are periodic and independent across multiplication, any correct analysis of cycles will produce the same final result 8.

Why Other Options Are Wrong:
Options 2, 4, 6, and 0 may arise if the cycles are misapplied or if one mistakenly truncates the analysis after fewer factors. For example, stopping after three factors might give 8 and then incorrectly adjusting. Zero would require at least one factor to be divisible by 10, which is not the case here because none of the base numbers end with 0 or 5 in combination with an even exponent to create a factor of 10.

Common Pitfalls:
Some learners forget to reduce the base to its unit digit before analyzing powers. Others miscalculate exponents modulo the cycle length, such as using 413 mod 2 instead of mod 4 for the digit 7. Confusing cycles for different last digits is also common. Careful stepwise handling of each base and exponent combination avoids these errors.

Final Answer:
The unit digit of the product (217)^413 × (819)^547 × (414)^624 × (342)^812 is 8.