Lecture Notes on Remainders by Raheep Prakash

Introduction to Remainders

• Importance of Remainders in exams (especially CAT)
  • Expect at least one question on Remainders in every exam
  • Remainders are a hyped topic in quantitative sections

Key Points about Remainders

• Properties of Remainders
  • The denominator can be used multiple times.
  • Example: When dividing 72 by 10, the remainder is 2.

Long Division Method for Remainders

• When dividing a number by a divisor, stop when the result is less than the divisor.
• Example: For 8 divided by 1, 2, 3, 4, 5, 6, 7, keep dividing until you get a result less than 8 for the remainder.

Splitting Numbers for Easier Calculation

• Using Denominators Multiple Times
  • Example: 132 divided by 10
    • Direct division gives remainder 2.
    • Splitting: (11 * 12) divided by 10 gives same answer.
• More Examples
  • Example: 504 divided by 5 gives remainder 4.
  • If split (7 * 8 * 9) divided by 5, repeat the process to confirm.*

Concept of Negative Remainders

• Negative remainders are used for convenience.
• Example: 8 divided by 10
  • Positive remainder: 8
  • Negative remainder: -2 (since 8 is 2 less than 10)
• Use the smaller magnitude for simplification.

Additional Examples of Negative Remainders

• Example: 12 divided by 13
  • Positive remainder: 12
  • Negative remainder: -1.
• Example: 37 divided by 8
  • Positive remainder: 5
  • Negative remainder: -3.

Combining Concepts

• Use both positive and negative remainders together to solve complex problems.
• Example: 18 * 19 * 20 * 21 divided by 17.
  • Calculate each part, using negative remainders if beneficial.*

Sample Problem and Solution

• Problem: 14 * 15 * 16 * 17 * 18 divided by 19
  • Use negative remainders where appropriate for easier calculation.

Practical Applications of Remainders

• Example: 38 raised to 138 divided by 39
  • Remainder is +1 (even power).
• Example: 3 raised to 196 divided by 4
  • Remainder is 1 (even power).

Summary

• Understanding how to manipulate remainders is crucial for solving quantitative problems efficiently.
• Next lessons will cover more applications and complex problems.

Important Terms

• Positive Remainder: The remainder when the division result is positive.
• Negative Remainder: The corresponding negative value for positive remainder.