Unit Digit of Powers

🎯 Learning Objective

Find the last digit of huge powers like 7^95 using cyclicity.

💡 Concept

• Unit digits repeat in cycles (cyclicity)
• Cycle of 2: 2,4,8,6 → repeats every 4
• Cycle of 3: 3,9,7,1 | Cycle of 7: 7,9,3,1 | Cycle of 8: 8,4,2,6
• 0,1,5,6 → unit digit never changes
• 4 & 9 have cycle of 2: 4,6 and 9,1
• Method: divide the power by 4, use the remainder to pick from the cycle

🧮 Key Formulas

Power ÷ 4 → remainder r → pick r-th term of cycle (r = 0 means 4th term)

✏️ Easy Example

Q. Find the unit digit of 2^15.

1. Cycle of 2 = (2,4,8,6)
2. 15 ÷ 4 → remainder 3
3. 3rd term of cycle = 8

Answer: 8

🇮🇳 Real-Life Example

Odometer last digits, dice game patterns — repeating cycles are everywhere; maths just names it cyclicity.

📝 Exam-Level Example

Q. Find the unit digit of 7^95.

1. Cycle of 7 = (7,9,3,1)
2. 95 ÷ 4 → remainder 3
3. 3rd term = 3

Answer: 3

📝 Exam-Level Example

Q. Unit digit of 123 × 587 × 987?

1. Take unit digits only: 3 × 7 × 7
2. 3×7 = 21 → 1; 1×7 = 7

Answer: 7

🪄 Memory Trick

Remember only two cycles: 2→2486 and 3→3971. 7's cycle is 3's cycle reversed (7931), 8's is 2's reversed (8426).

⚠️ Common Mistakes

• ❌ Using remainder 0 as position 0 (it means the 4th/last term)
• ❌ Multiplying full numbers instead of unit digits

🏆 Exam Tips

• ✅ In products, keep only unit digits at every step
• ✅ Cyclicity questions are 15-second questions once practiced

📌 Summary

• Unit digits repeat every 4 (max)
• 0,1,5,6 never change; 4,9 have cycle 2
• Power ÷ 4, remainder decides; remainder 0 → 4th term
• Products → multiply unit digits only