Chapter Summary: Prime Numbers and Factors

Key Concepts

• Prime Numbers: Numbers that have only two factors: 1 and themselves (e.g., 2, 3, 5, 7, 11).
• Composite Numbers: Numbers that have more than two factors (e.g., 4, 6, 8, 9).
• Factors: A number is a factor of another if it divides the other number without leaving a remainder.
• Co-prime Numbers: Two numbers that have no common factors other than 1.
• Prime Factorisation: Every number greater than 1 can be expressed as a product of prime numbers.

Examples

• Prime Factorisation Examples:
  • 64 = 2 x 2 x 2 x 2 x 2 x 2
  • 105 = 3 x 5 x 7
  • 198 = 2 x 3 x 3 x 11

Important Questions

• Which of the following numbers are prime: 23, 51, 37, 26? (Answer: 23 and 37)
• Find seven consecutive composite numbers between 1 and 100. (Answer: 90, 91, 92, 93, 94, 95, 96)
• What is the smallest number whose prime factorisation has three different prime numbers? (Answer: 30)

Common Mistakes

• Confusing prime numbers with composite numbers.
• Misidentifying factors and multiples.

Exam Tips

• Always check the definition of prime and composite numbers before answering questions.
• Use prime factorisation to determine factors and co-primality.

Chapter 5 - Solutions

Prime Time

Common Multiples and Common Factors

• Idli-Vada Game: Children sit in a circle and play a game of numbers.
  • Players say 'idli' for multiples of 3 (e.g., 3, 6, 9, 12, 18).
  • Players say 'vada' for multiples of 5 (e.g., 5, 10, 15, 20).
  • For numbers that are multiples of both 3 and 5, players say 'idli-vada' (first instance is 15).

Prime Factorization

• Examples:
  • 64 = 2 x 2 x 2 x 2 x 2 x 2
  • 104 = 2 x 2 x 2 x 13
  • 105 = 3 x 5 x 7
  • 243 = 3 x 3 x 3 x 3 x 3
  • 320 = 2 x 2 x 2 x 2 x 2 x 5

Co-prime Numbers

• Two numbers are co-prime if they have no common factors other than 1.

Prime and Composite Numbers

• Prime Numbers: Numbers like 2, 3, 5, 7, 11 (only two factors: 1 and themselves).
• Composite Numbers: Numbers like 4, 6, 8, 9 (more than two factors).

Common Mistakes

• Confusing prime numbers with composite numbers.
• Misidentifying common multiples.

Tips

• Always check the prime factorization to determine if numbers are co-prime.
• Use the idli-vada game to practice identifying multiples.

Common Mistakes and Exam Tips

Common Pitfalls

• Misunderstanding the Game Rules: Students may confuse when to say 'idli', 'vada', or 'idli-vada'. It's crucial to remember that 'idli' is for multiples of 3, 'vada' for multiples of 5, and 'idli-vada' for multiples of both.
• Counting Errors: When counting how many times 'idli', 'vada', or 'idli-vada' is said, students might miscount, especially in larger ranges. Always double-check your counts.
• Identifying Co-prime Numbers: Students often incorrectly identify co-prime numbers. Ensure to check for common factors thoroughly, not just by one method of factorization.
• Assuming All Multiples are Equal: Some may think that all multiples of a number are the same. For example, multiples of 3 and 5 will have overlaps, and students should identify these overlaps correctly.

Tips for Success

• Practice with Examples: Regularly practice with different sets of numbers to become familiar with identifying multiples and co-prime pairs.
• Use Factorization: When determining if two numbers are co-prime, always use prime factorization to check for common factors.
• Visual Aids: Utilize diagrams like Venn diagrams to visualize common multiples and factors, which can help in understanding relationships between numbers.
• Double-Check Work: After solving problems, go back and verify your answers, especially in counting scenarios or when determining co-primality.