Remainders
A remainder is the amount left over after division when one number does not divide another exactly. Remainder problems are frequently asked in aptitude, banking, SSC, Loksewa, and entrance examinations.
Fundamental Principles
Remainder
The value left after dividing a number by another number.
Dividend
The number being divided.
Divisor
The number by which division is performed.
Essential Formulation Tips
- Remainder is always less than the divisor.
- Use modular arithmetic for large powers.
- Look for repeating patterns in units digits.
- Reduce large numbers before performing calculations.
- Apply divisibility rules whenever possible.
Shortcut Execution Techniques
- Dividend = Divisor × Quotient + Remainder
- For addition: add remainders and divide again if needed.
- For multiplication: multiply remainders and divide again if needed.
- Powers often follow cyclic patterns.
- Use modular arithmetic to simplify large calculations.
Contextual Inquiries (FAQs)
Q: Can a remainder be greater than the divisor?
A: No. A remainder is always smaller than the divisor.
Q: What is the remainder when a number is exactly divisible?
A: The remainder is 0.
Q: Why are remainder problems important?
A: They help solve large-number calculations quickly using patterns and modular arithmetic.
Example Breakdown: Basic Remainder
Most basic remainder problem.25 = 4 × 6 + 1
Remainder = 1
Example Breakdown: Remainder of a Sum
Useful shortcut.17 leaves remainder 2 when divided by 5
23 leaves remainder 3 when divided by 5
2 + 3 = 5
Remainder = 0
Example Breakdown: Remainder of a Power
Common aptitude pattern.2 ≡ 2 (mod 3)
2² ≡ 1 (mod 3)
2⁵ = 2⁴ × 2
1 × 2 = 2
Remainder = 2
Set 1: Introduction to Remainders
Understanding basic division and the remainder concept.
Q1. What is the remainder when 7 is divided by 3?
Q2. What is the remainder when 10 is divided by 5?
Q3. If a number is perfectly divisible, the remainder is:
Q4. What is the remainder when 13 is divided by 4?
Q5. Which of these yields a remainder of 0 when divided by 2?
Q6. What is the remainder when 20 is divided by 3?
Q7. The remainder must always be ____ the divisor.
Q8. What is the remainder when 5 is divided by 6?
Q9. What is the remainder when 18 is divided by 6?
Q10. What is the remainder when 25 is divided by 4?