Remainder Concepts

🎯 Learning Objective

Use Dividend = Divisor × Quotient + Remainder and solve remainder questions fast.

💡 Concept

• Dividend = Divisor × Quotient + Remainder
• Remainder is always smaller than the divisor
• Successive division: divide, then divide the quotient again
• Negative remainder shortcut: 15 ÷ 8 → remainder 7, or think −1 (7 = 8−1) for big powers

🧮 Key Formulas

Dividend = Divisor × Quotient + Remainder

>

0 ≤ Remainder < Divisor

✏️ Easy Example

Q. A number when divided by 7 gives quotient 12 and remainder 4. Find the number.

1. Number = 7 × 12 + 4
2. = 84 + 4

Answer: 88

🇮🇳 Real-Life Example

112 laddoos packed in boxes of 9: 12 boxes and 4 laddoos left over — that leftover is the remainder.

📝 Exam-Level Example

Q. Find the remainder when 2^10 is divided by 3.

1. 2 ÷ 3 → remainder is −1 (shortcut)
2. So 2^10 → (−1)^10 = +1

Answer: 1

📝 Exam-Level Example

Q. The largest number that divides 70 and 125 leaving remainders 5 and 8?

1. Subtract remainders: 70−5 = 65, 125−8 = 117
2. Required number = HCF(65, 117)
3. 65 = 5×13, 117 = 9×13 → HCF = 13

Answer: 13

🪄 Memory Trick

When 'same remainder' questions appear, subtract the remainder first, then find HCF.

⚠️ Common Mistakes

• ❌ Writing remainder ≥ divisor (impossible)
• ❌ Forgetting to subtract remainders before HCF

🏆 Exam Tips

• ✅ Learn powers of 2, 3, 7 up to 5–6 terms
• ✅ Negative remainder trick saves 30+ seconds on power questions

📌 Summary

• Dividend = Divisor × Quotient + Remainder
• Remainder < Divisor always
• Power questions → use ±1 remainder pattern
• 'Leaves remainder' + 'largest divisor' → subtract then HCF