Summary of "Lecture-09 || Number System Part-09 (Remainder-02)"

Summary of Lecture-09 || Number System Part-09 (Remainder-02)

Main Ideas and Concepts Covered

Review of Remainder Basics (Part 1 Recap):
- Understanding what a remainder is.
- How to find remainders, especially with powers.
- Using powers that yield remainders of ±1 to simplify calculations.
Handling Common Factors in Dividend and Divisor:
- When dividend and divisor share a common factor, simplify by dividing both by that factor.
- After finding the remainder with simplified numbers, multiply the remainder by the common factor to get the final remainder.
- Example: For (80 \div 24), dividing numerator and denominator by 8 simplifies the problem; multiply the final remainder by 8.
Example Problems Involving Powers and Divisors with Common Factors:
- Example 1: (5^{150} \div 150)
  - Factor 150 as (2 \times 3 \times 5^2).
  - Simplify powers accordingly and multiply back by (5^2 = 25) to get the correct remainder.
- Example 2: (2^{96} \div 96)
  - Factor 96 as (2^5 \times 3).
  - Use remainder properties of powers and multiply back by (2^5 = 32) for the final positive remainder.
Important Note on Remainders:
- Remainders are always non-negative.
- If a negative remainder arises, add the divisor to get the positive remainder.
Remainder Problems Involving Linear Expressions:
- If a number (n) leaves remainder (r) when divided by (d), then: [ n = d \times x + r ]
- To find the remainder when (n) is divided by another number, substitute and simplify accordingly.
- Example: A number leaves remainder 19 on division by 119. Find remainder when divided by 17 by expressing (n = 119x + 19) and reducing modulo 17.
Key Theorems on Powers and Divisibility:
- (x^n - y^n) is divisible by (x - y) for all natural numbers (n).
- (x^n + y^n) is divisible by (x + y) if (n) is odd.
- When dividing (x^n - y^n) by (x + y), divisibility depends on (n) being even.
- Examples were provided to illustrate these divisibility rules.
Summary of Three Important Points:
- (x^n - y^n) divisible by (x - y) for any natural (n).
- (x^n - y^n) divisible by (x + y) if (n) is even.
- (x^n + y^n) divisible by (x + y) if (n) is odd.
Successive (Sequential) Remainder Concept:
- When a number is divided sequentially by two divisors, the remainders at each step relate to each other.
- For example, dividing a number by 5 gives remainder 2, then dividing the quotient by 7 gives remainder 4.
- Use the relations: [ n = 5q_1 + 2, \quad q_1 = 7q_2 + 4 ]
- Shortcut method introduced to quickly find the combined remainder when dividing by the product of divisors (like 35 in the example):
  - Write zeros and remainders in a column, multiply diagonally, add, and get the combined remainder.
  - General form: [ n = \text{(combined remainder)} + \text{(product of divisors)} \times m ] where (m) is an integer.
Extension to Multiple Divisors:
- The same logic applies when dividing by three or more numbers sequentially.
- Use the shortcut method with multiplication and addition steps to find the general form of the number.
Practice Questions and Application: - Several examples from CDS (Combined Defence Services) exam papers were discussed. - Practice problems involving powers, remainders, and divisibility rules. - Emphasis on understanding concepts rather than rote calculation. - Encouragement to use shortcuts and logical reasoning for faster problem-solving.
Final Notes: - Remainder problems are common in competitive exams. - Practice sheets with 25 questions provided. - Next class and doubt sessions scheduled, including an extra class on Saturday. - Encouragement to attempt problems independently and ask doubts.

Detailed Methodologies and Instructions

Simplifying Remainder Problems with Common Factors

Identify common factor (k) in dividend and divisor.
Divide both dividend and divisor by (k).
Calculate remainder with simplified numbers.
Multiply the remainder by (k) to get the final remainder.

Finding Remainder for Powers

Factor the divisor into prime factors.
Use properties of powers modulo each factor.
Use cyclicity or powers that yield ±1 remainders for simplification.
Multiply back factors canceled earlier.

Divisibility Rules for Powers

(x^n - y^n) divisible by (x - y) for all natural (n).
(x^n - y^n) divisible by (x + y) if (n) is even.
(x^n + y^n) divisible by (x + y) if (n) is odd.

Successive Remainder Shortcut

Write divisors and their remainders in a vertical column, starting with zero at the top.
Multiply diagonally and add vertically to get combined remainder.
General form: [ n = (\text{combined remainder}) + (\text{product of divisors}) \times m ] where (m) is any integer.
Use this to find remainder when dividing by product of divisors.

Example of Successive Remainder

Dividing number by 5 and 7 gives remainders 2 and 4 respectively.
Construct: 0 4 4 × 5 = 20 20 + 2 = 22
Product of divisors = (5 \times 7 = 35).
General form: [ n = 22 + 35m ]
Remainder when dividing by 35 is 22.

Speakers/Sources Featured

Primary Speaker: The instructor/teacher conducting the lecture (name not explicitly mentioned, but addresses students like Abhishek, Amit, Satyam, etc.).
Students (mentioned during interaction):
- Pankaj
- Amit
- Abhishek
- Satyam
- Lakshya
- Saurabh
- Tekchand
- Aftab
- Govind
- Yush
- Ovesh
- Suraj
- Tomar
- Deepak
- Anuraj
- Rao Saheb
- Diwakar
- Judge Ji