Units digit evaluation — large exponent Determine the units digit (ones place) of (6817)^754.

Introduction / Context:
For huge exponents, only the base’s last digit affects the final units digit. Recognizing the repeating cycle of units digits for powers of a digit greatly simplifies computation. Here, 6817 ends with 7, so we analyze powers of 7 modulo 10.

Given Data / Assumptions:

• We need the units digit of 6817^754.
• Only the last digit 7 matters for units digit analysis.
• Cycles of powers modulo 10 repeat with small periods.

Concept / Approach:
Powers of 7 mod 10 cycle as: 7, 9, 3, 1, and then repeat every 4. Therefore, we compute the exponent modulo 4 to select the correct position in the cycle.

Step-by-Step Solution:
Cycle for 7^k mod 10: [7, 9, 3, 1] with period 4.Compute 754 mod 4: since 4*188 = 752, remainder = 2.Remainder 2 corresponds to the second element in the cycle, which is 9.Thus, the units digit is 9.

Verification / Alternative check:
Confirm smaller examples: 7^1 → 7, 7^2 → 49 (ends 9), 7^3 → 343 (ends 3), 7^4 → 2401 (ends 1), and repeats. The mapping of remainders 1–4 to [7,9,3,1] is consistent.

Why Other Options Are Wrong:
8, 4, 2 correspond to other digit cycles, not the 7^k cycle at remainder 2.

Common Pitfalls:
Using the exponent itself instead of its remainder modulo the cycle length; mis-ordering the cycle; or assuming every digit has the same period length.

Final Answer:
9