Units digit of a sum with large exponents Find the units digit of 3^65 + 6^59 + 7^71.

Introduction / Context:
To find the units digit of a sum involving large exponents, we compute each term’s units digit independently using known cycles, then add and reduce modulo 10. This approach avoids dealing with massive numbers.

Given Data / Assumptions:

• Expression: 3^65 + 6^59 + 7^71.
• Only last digits matter: 3, 6, 7.
• Each base has a small repeating cycle modulo 10.

Concept / Approach:
Cycles: 3^k mod 10 → [3, 9, 7, 1] (period 4). 6^k mod 10 → always 6. 7^k mod 10 → [7, 9, 3, 1] (period 4). Compute exponents modulo 4 where applicable, then add resulting units digits and reduce modulo 10.

Step-by-Step Solution:
For 3^65: 65 mod 4 = 1 → units digit 3.For 6^59: always 6 → units digit 6.For 7^71: 71 mod 4 = 3 → units digit 3.Sum of units digits: 3 + 6 + 3 = 12 → 12 mod 10 = 2.

Verification / Alternative check:
Quick sanity: 3-cycle position 1 gives 3; 7-cycle position 3 gives 3; 6 is constant. The final 2 is consistent with common results for such mixed-base problems.

Why Other Options Are Wrong:
1, 4, 6 arise from selecting the wrong cycle position or forgetting to reduce the final sum modulo 10.

Common Pitfalls:
Forgetting that 6^n ends in 6 for all n ≥ 1; computing 65 mod 4 incorrectly; adding the units digits but not reducing modulo 10 at the end.

Final Answer:
2