Chapter 4: Number Theory

§ 4.1 Factors and Multiples

Topics

• What is number theory?
  The study of patterns in whole numbers. `{2,3,5,7,11, ...}`
• Factor
  4 is a factor of 12.
  
• Multiple
  12 is a multiple of 3.
  
• Divides:
  `a|b` if `a*n=b`
  a divides b if there is a natural number such that a times n equals b.
• Models:
  linear
  
  rectangular
  
• Use color rods as linear model for factors and multiples:
  
  Color Rods App
• Use color tiles as a rectangular model for factors and multiples:
  
  Color Tiles App
• Even and Odd
• Divisibility Tests

  2

  the ones digit is a `0,2,4,6,8`

  3

  the sum of the digit is a multiple of 3

  4

  the two right-most digits form a multiple of 4

  5

  the ones digit is a `0,5`

  6

  meets the tests for 2 and 3

  8

  the three right-most digits form a multiple of 8

  9

  the sum of the digit is a multiple of 9

  10

  the ones digit is a `0`

  11

  the difference between the sum of the digits from the even powers of 10 and the sum of the digits from the odd powers of 10, is a multiple of 3

• Divisibility Properties:
  • If `a|b`, then `a|(n*b)`,
  • If `a|b` and `a|c`, then `a|(b+c)`,
  • If `a|b` and `a cancel(|)c`, then `a cancel(|)(b+c)`
• Prime and Composite
  Play with rectangular arrays
• Factor Trees
• Prime number test
  Suppose n is a whole number and k is the smallest whole number such that k × k is greater than n. If there is no prime number less than k that is a factor of n, then n is a prime number.
• Sieve of Eratosthenes