Divisibility Rules (2 to 11)

🎯 Learning Objective

Check divisibility of any number by 2–11 in seconds without actual division.

💡 Concept

• By 2: last digit even (0,2,4,6,8)
• By 3: sum of digits divisible by 3
• By 4: last 2 digits divisible by 4
• By 5: last digit 0 or 5
• By 6: divisible by both 2 and 3
• By 8: last 3 digits divisible by 8
• By 9: digit sum divisible by 9
• By 11: (sum of odd-place digits) − (sum of even-place digits) = 0 or multiple of 11

✏️ Easy Example

Q. Is 4,572 divisible by 9?

1. Digit sum = 4 + 5 + 7 + 2 = 18
2. 18 ÷ 9 = 2, so yes

Answer: Yes, divisible by 9

🇮🇳 Real-Life Example

Splitting a ₹4,572 bill equally among 9 friends? The digit-sum rule tells you instantly it divides perfectly — ₹508 each.

📝 Exam-Level Example

Q. Find the smallest value of * so that 63*57 is divisible by 11.

1. Odd places (from right): 7, *, 6 → sum = 13 + *
2. Even places: 5, 3 → sum = 8
3. Difference = (13 + *) − 8 = 5 + *
4. 5 + * must be 11 → * = 6

Answer: 6

🪄 Memory Trick

For 6 = check 2 AND 3. For 12 = check 3 AND 4. Break composite divisors into co-prime parts.

⚠️ Common Mistakes

• ❌ Checking 4 with the last digit only (needs last TWO digits)
• ❌ For 11, adding all digits instead of alternating sums

🏆 Exam Tips

• ✅ Digit-sum questions (3 & 9) are guaranteed in RRB exams
• ✅ Practice till each check takes under 10 seconds

📌 Summary

• 2/5/10 → last digit; 4 → last 2 digits; 8 → last 3 digits
• 3 & 9 → digit sum
• 11 → alternate sum difference
• Composite divisor → split into co-prime factors