Unit digit of 3⁶⁵ × 6⁵⁹ × 7⁷¹ — Aptitude Question
Unit digit of 3⁶⁵ × 6⁵⁹ × 7⁷¹: What is the unit digit of 3⁶⁵ × 6⁵⁹ × 7⁷¹? Answer and worked solution on CodeKairo.
- Topic: Number System
- Section: Quantitative Aptitude
- Difficulty: Medium
- Time target: 75 seconds
Question
What is the unit digit of 3⁶⁵ × 6⁵⁹ × 7⁷¹?
Options
- 2
- 4
- 6
- 8
Show the answer and the worked solution
Correct answer: B. 4
Worked solution
- 3ⁿ cycles 3, 9, 7, 1; 65 mod 4 = 1 → unit digit 3.
- 6ⁿ always ends in 6.
- 7ⁿ cycles 7, 9, 3, 1; 71 mod 4 = 3 → unit digit 3.
- 3 × 6 = 18 → 8; 8 × 3 = 24 → 4.
Answer: 4.
The idea
Multiply only the unit digits, reducing to a single digit after each step. Remember the fixed points: 0, 1, 5 and 6 always end in themselves, whatever the power.
Shortcut. Pair 3⁶⁵ with 7⁶⁵ to make 21⁶⁵, which ends in 1. What remains is 6⁵⁹ × 7⁶, and 6 × 9 = 54 ends in 4.
Common trap. 8 — the unit digit of 3 × 6 = 18, as if 7⁷¹ ended in 1; 71 mod 4 = 3, so it ends in 3 and 8 × 3 = 24 gives 4.