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Structures 5

Structures 5. Number Theory. Using the structure of numbers to form arguments. odds and evens multiples divisibility. The sum of two odd numbers is even. (2n + 1) + (2m + 1) = 2n + 2m + 2 = 2(n + m + 1). 2n + 1 represents “odd number ”. 2( … ) is even.

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Structures 5

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  1. Structures 5 Number Theory

  2. Using the structure of numbers to form arguments • odds and evens • multiples • divisibility

  3. The sum of two odd numbers is even (2n + 1) + (2m + 1) = 2n + 2m + 2 = 2(n + m + 1) 2n + 1 represents “odd number” 2( … ) is even

  4. The sum of 3 consecutive numbers is divisible by 3. What does “divisible by 3” look like? How can you represent “consecutive numbers”?

  5. Is it true in general that the sum of kconsecutive numbers is divisible by k? Form and prove conjectures.

  6. Primes • prime factorisation • how many prime numbers are there?

  7. Primes and Factorisation 24 12 2 6 2 3 2 factor tree for 24

  8. Primes and Factorisation 24 12 2 6 2 3 2

  9. Primes and Factorisation 24 24 12 2 6 4 6 2 3 2 2 2 3 2

  10. How many different factor trees? • How many different factors? Given a number expressed as a product of primes, how many different factors does it have? e.g. Consider the numbers 1 – 100. Which numbers have 1, 2, 3, 4, etc. different factors?

  11. Formulae for prime numbers? Mersenne numbers

  12. How many prime numbers are there?

  13. Approximations for the number of primes less than x

  14. Is there a largest prime number?

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