Divisibility: From-To and Between Cases

Jul 23, 2024

Lecture Notes on Divisibility

Introduction and Changes to Class Plan

  • Last class was canceled due to changes in the number of questions and content to ensure clearer understanding.
  • The updated sheet includes added and selective questions for better clarity.
  • New question numbers and changes will be reflected in the uploaded sheets.
  • Focus on divisibility rules and types of questions to ensure thorough understanding.

Key Points from the Lecture

Divisibility Concepts

  • Divisibility Basics: Understanding how numbers are divisible by other numbers using rules and patterns.
  • Types of Questions: Two main types include 'from-to' and 'between' cases.

'From-To' Cases vs. 'Between' Cases

  • From-To Cases: Involves all numbers from the start to the end. For example, 'From 1 to 100' includes 1, 2, ..., 100.
    • Number of terms formula: ((Last term - First term) / Difference) + 1.
  • Between Cases: Does not include the start and end numbers. For example, 'Between 1 and 10' includes 2, 3, ..., 9.
    • Number of terms formula: ((Last term - First term) / Difference) - 1.

Examples and Calculations

  • From 200 to 300: Number of natural numbers is calculated as (300 - 200) / 1 + 1 = 101.
  • Between 30 and 50: Number of natural numbers is calculated as (49 - 31) / 1 + 1 = 19.

Divisibility by a Single Number

  • Example: Count how many numbers are divisible by 6 from 1 to 3000.
    • Use multiples of 6, 12, 18,...3000.
    • Number of such numbers: 3000 / 6 = 500.

Divisibility by Multiple Numbers

  • LCM Method: For numbers divisible by multiple numbers, calculate the LCM first and then find the divisible numbers.
  • Example: Numbers divisible by 3 and 5 from 1 to 250:
    • LCM of 3 and 5 is 15.
    • Number of terms: 250 / 15 = 16.

More Complex Cases

  • Multiple Intervals: When calculating between intervals like 100 to 3000:
    • Consider LCM and begin from both extremes (1 to 3000 and 1 to 99), and subtract the smaller range’s result from the larger.
    • 100 to 3000 divisible by 6: (3000 / 6) - (99 / 6) = 484.

Practice Questions

  • Example Problems: Various problems involving finding numbers divisible by different combinations within specified ranges.
    • e.g., dividing numbers from 1 to 1000 by 4, 5, and 7.
    • Further batches involve breaking complex problems into simpler calculations.

Key Takeaways and Further Work

  • Comprehensive approach ensures clear understanding, stepping from simple to complex problems progressively.
  • Acknowledged content coverage might expand to multiple classes for detailed comprehension.
  • Upcoming Topics: More on 'between' cases and 'not divisible' questions to ensure all aspects are covered sufficiently.
  • Homework: Review the practice problems provided and ensure strong grip on both 'from-to' and 'between' problem types.

Stay tuned for the next class: We will cover the divisibility for 'between' cases, moving then to 'not divisible' scenarios followed by special types.

End of Notes