number theory

Newton School of Technology Mathematics Notes

Number Theory Overview

  • Course Title: Number Theory

    • Instructor: Rishabh Bafna

    • CSA 102 Mathematics

Quick Revision

  • Previous Topics Reviewed:

    • Percentages

    • Ratio and Proportion

    • Mixtures and Allegations

Profit Calculation Example

  • Example: Dealer sells sugar at different profits.

    • Total sugar: 1000 kg

    • Profit: 8% and 18% with overall profit 14%

    • Goal: Find quantity sold at 18% profit.

Percentage Change Examples

Increase in Stock Price

  • Question: Stock price increased from Rs. 50 to Rs. 75

    • Calculation:

      % change = (New Value - Original Value) / Original Value x 100

      • = (75-50) / 50 x 100 = 50% increase

Decrease in Stock Price

  • Question: Stock price decreased from Rs. 75 to Rs. 50

    • Calculation:

    • % change = (New Value - Original Value) / Original Value x 100

    • = (50-75) / 75 x 100 = -33.33% decrease

Basics of Number Theory

  • Definition: Number Theory studies numbers and their relationships.

  • Skill Focus: Shortcut seeking and efficiency in calculations.

Cyclical Patterns Example

  • Question: Given a pattern with three colors, find color for 8 × 17.

    • Calculation:

    • 8 × 17 = 136

    • 136 % 3 = 1

    • Answer: Blue (Remainder 1 corresponds to Blue)

Divisibility Rules

  • Divisibility by a number means no remainder when divided.

    • Example: 12 is divisible by 4 (no remainder).

    • Not Divisible Example: 12 is not divisible by 5 (has remainder).

Practical Divisibility Application

  • Example: Five-digit number ending in 0.

    • Divisibility:

      • By 2: Yes

      • By 4: Indeterminate

      • By 5: Yes

Specific Divisibility Rules

  • By 2: Last digit must be even.

  • By 5: Last digit must be 0 or 5.

  • By 10: Last digit must be 0.

  • By 4: Last two digits must form a number divisible by 4.

Analyzing Divisibility by 4

  • Check divisibility using last two digits, e.g., 234.

  • Analyze sets of digits and their remainders after division by 4.

Summarized Divisibility Rules

  • Any whole number assessed from its digits rolled up into sums will lead to divisibility by the respective bases (2, 3, 5, etc.).

Prime Factorization

  • Definition: Breakdown of a number into its prime factors.

    • Example: Factorization of 40 = 2 × 2 × 2 × 5

  • Fundamental Theorem: Each number has a unique prime factorization.

Finding Number of Divisors

  • Process: Calculate using prime factorization.

  • Formula: If n = p1^k1 × p2^k2 × ... × pm^km, then the number of divisors d(n) = (k1 + 1)(k2 + 1)...(km + 1).

Practical Application Example

  • Example: 300 slices pizza, determining group sizes for even distribution.

    • Prime factorization gives possible divisor combinations.

Problem-solving Example: Doors Scenario

  • A hallway with 100 doors with various patterns of changing states.

  • Track changes per person as they toggled doors at multiples position.

  • Example: How many times door no. 88 is changed?

Key Takeaways

  • Mastered:

    • Divisibility rules for numbers.

    • Prime factorization methods and applications in real-world problems.

Quiz Time

  • Engage in quizzes to reinforce learning.

Conclusion

  • Summary of major principles in number theory and arithmetic thinking for problem-solving and practical applications.