Summary of "Lecture-08 || Number System Part-08 (Consecutive Integer)"
Summary of Lecture-08 || Number System Part-08 (Consecutive Integer)
Main Topic: Remainders (Reminders) in Division
This lecture focuses on the concept of remainders in division problems, explaining fundamental properties, common mistakes, and advanced applications including powers and modular arithmetic. The content is delivered interactively with examples, student engagement, and problem-solving exercises.
Key Concepts and Lessons
1. Basic Definition of Remainder
- When a number (dividend) is divided by another number (divisor), the amount left over after division is called the remainder.
- Example: 19 divided by 5 → Quotient = 3, Remainder = 4.
- Formula: Dividend = Divisor × Quotient + Remainder
2. Terminology
- Dividend (Bhajya in Hindi): Number being divided.
- Divisor (Bhajaka in Hindi): Number by which division is done.
- Quotient (Bhaajak): Number of complete groups.
- Remainder (Shesh): Leftover part after division.
3. Properties of Remainders
- Remainder is always a whole number (integer).
- Remainder is always less than the divisor.
- Remainder can be zero.
- When numerator < denominator in a division, numerator itself is the remainder.
4. Common Mistakes
- Incorrect cancellation in division problems (e.g., reducing terms incorrectly).
- Misunderstanding remainder when numerator is smaller than denominator.
5. Remainder in Expressions Involving Multiplication and Addition
- Remainders follow similar additive, subtractive, and multiplicative properties as unit digits.
- For example:
- If remainder of (n_1) divided by (a) is (r_1), and remainder of (n_2) divided by (a) is (r_2), then:
- Remainder of ((n_1 + n_2)) divided by (a) is ((r_1 + r_2) \bmod a).
- Remainder of ((n_1 \times n_2)) divided by (a) is ((r_1 \times r_2) \bmod a).
- Remainder of ((n_1 - n_2)) divided by (a) is ((r_1 - r_2) \bmod a), with special handling for negative remainders.
- If remainder of (n_1) divided by (a) is (r_1), and remainder of (n_2) divided by (a) is (r_2), then:
6. Handling Negative Remainders
- Remainders are by default non-negative.
- Negative remainders can be used conceptually to simplify calculations.
- If remainder is negative, add the divisor to get the positive equivalent. Example: If remainder = -2 when dividing by 11, actual remainder = (11 - 2 = 9).
- Negative remainder interpretation helps in easier mental calculations.
7. Remainders with Exponents (Modular Exponentiation)
- When dealing with powers (e.g., (a^b \div c)), find the remainder by:
- Identifying the pattern or cycle of remainders of powers of (a) modulo (c).
- Grouping powers based on cycle length.
- Using the fact that (a^{\text{cycle_length}} \equiv 1 \pmod{c}).
- Example: (2^{50} \div 3): Powers of 2 mod 3 cycle every 2 powers: (2^1 \equiv 2), (2^2 \equiv 1), so group into 25 cycles of 2. Result remainder = 1.
- The method involves finding the smallest power (k) such that (a^k \equiv 1 \pmod{c}).
8. Examples of Powers and Remainders
- (3^{48} \div 8), (2^{52} \div 7), (5^{69} \div 24), (2^{70} \div 11), etc.
- Calculation involves identifying cycles and leftover powers.
9. Format Theorem (Fermat’s Little Theorem)
- A shortcut to solve remainder problems with powers when divisor is prime.
- If (p) is prime and (a) is coprime to (p), then: [ a^{p-1} \equiv 1 \pmod{p} ]
- This means for (a^p \div p), remainder is 1 if conditions hold.
- Examples: (2^{10} \div 11), (3^{18} \div 19), (2^{100} \div 101), (8^{46} \div 47).
- The theorem helps solve large power remainder problems quickly.
- Note: Not applicable if divisor is not prime or (a) and divisor are not coprime.
10. Negative Remainder in Powers
- Powers can also have negative remainders conceptually.
- Example: (7^2 \div 50) remainder is -1 (or 49).
- Final answers should always be given as positive remainders.
Methodology / Instructions for Solving Remainder Problems
- Understand the division problem and identify dividend, divisor, quotient, and remainder.
- Use the formula: [ \text{Dividend} = \text{Divisor} \times \text{Quotient} + \text{Remainder} ]
- For powers, find the cycle length of the base modulo divisor.
- Group the exponent by cycle length and find remainder of leftover exponent.
- Use Format Theorem for prime divisors and coprime bases to shortcut calculations.
- Handle negative remainders by adding the divisor to get positive remainder.
- Apply additive and multiplicative properties of remainders for combined expressions.
- Verify answers through example problems and avoid common mistakes like incorrect cancellation.
Examples Covered
- Simple division remainder problems (e.g., 19 ÷ 5, 47 ÷ 5).
- Dividing numbers with remainder conditions (e.g., divisor is 4 times remainder).
- Modular arithmetic with sums and differences of remainders.
- Powers with modular division (e.g., (2^{50} \div 3), (3^{48} \div 8), (2^{70} \div 11)).
- Using Format Theorem for prime divisors.
- Negative remainder interpretation in division and powers.
Important Notes
- Always write remainder as a non-negative integer in final answers.
- Negative remainders are a conceptual tool for simplifying calculations.
- Format Theorem is a powerful shortcut but only applies under specific conditions (prime divisor, coprime base).
- Larger modular arithmetic theorems like Wilson’s, Euler’s, and Chinese Remainder Theorem exist but are beyond the current syllabus.
- Practice and revision are essential for mastering remainder problems.
Speakers / Sources Featured
- Primary Speaker: The instructor/teacher conducting the lecture (name not explicitly mentioned).
- Students/Participants Mentioned: Laksh, Diwakar, Mark, Abhishek, Saurabh, Satyam, Tek Chand, Umesh, Vivek, Aftab, Govind Pandit, Osh, Satya, Priyanshu, Ovesh, Amitraj, Ashraf, Bala Krishna, Mayank, Ratu, KGS user, Anshu.
This summary encapsulates the main ideas, methodologies, and examples discussed in the lecture on remainders and modular arithmetic, providing a comprehensive guide to the topic.
Category
Educational
