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Chapter 3. Elementary Number Theory and Methods of Proof. 3.3. Direct Proof and Counterexample 3 Divisibility. Divisibility. Definition If n and d are integers, then n is divisible by d if, and only if, n = dk for some integers k .

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chapter 3

Chapter 3

Elementary Number Theory and Methods of Proof

slide2

3.3

Direct Proof and Counterexample 3

Divisibility

divisibility
Divisibility
  • Definition
    • If n and d are integers, then n is divisible by d if, and only if, n = dk for some integers k.
    • The notation d|n is read “d divides n.”
    • Symbolically, if n and d are integers,
    • d|n ⇔ ∃an integer k such that n = dk.
examples
Examples
  • Divisibility
    • Is 21 divisible by 3?
      • Yes, 21 = 3 * 7
    • Does 5 divide 40?
      • Yes, 40 = 5 * 8
    • Does 7|42?
      • Yes, 42 = 7 * 6
    • Is 32 a multiple of -16?
      • Yes, 32 = (-16) * (-2)
    • Is 7 a factor of -7?
      • Yes, -7 = 7 * (-1)
divisors
Divisors
  • Divisors of zero
    • If d is any integer, does d divide 0?
    • Recall: d|n ⇔ ∃an integer k such that n = dk
    • n = 0, is there an integer k such that dk = 0 (n)
    • Yes, 0 = d*0 (0 is an integer)
  • Divisors of one
    • Which integers divide 1?
    • 1 = dk, 1*1 or (-1) * (-1)
divisors1
Divisors
  • Positive Divisors of a Positive Number
    • Suppose a and b are positive integers and a|b.
    • Is a≤b?
      • Yes. a|b means that b = ka for some integer k. k must be positive because b and a are positive.
      • 1 ≤ k
      • a ≤ k * a = b, hence, a ≤ b
divisibility1
Divisibility
  • Algebraic expressions
    • 3a + 3b divisible by 3 (a & b are integers)?
    • Yes. By distributive law
      • 3a + 3b = 3(a + b), a & b are integers so the sum of a, b are integers.
    • 10km divisible by 5 (k & m are integers)?
      • 10km = 5 * (2km)
prime numbers
Prime Numbers
  • An alternative way to define a prime number is to say that an integer n > 1 is prime if, and only if, its only positive integer divisors are 1 and itself.
transitivity of divisibility
Transitivity of Divisibility
  • Prove that for all integers a, b, and c, if a|b and b|c, then a|c.
    • Starting point: Suppose a, b, and c are particular but arbitrarily chosen integers such that a|b and b|c.
    • To show: a|c
      • a|c, c = a*(some integers)
      • since a|b, b = ar for some integer r
      • And since b|c, c = bs
      • c = (ar)s(substitue for b)
      • c = a(rs) (assoc law)
      • c = ak (such that rs is integer due to close property of int)
  • Theorem 3.3.1 Transitivity of Divisibility
divisibility by a prime
Divisibility by a Prime
  • Any integer n > 1 is divisible by a prime number.
    • Suppose n is a integer that is greater than 1.
    • If n is prime, then n is divisible by a prime number (namely itself). If n is not prime, then
      • n = r0s0 where r0 and s0 are integers and 1 < r0 < n and 1 < s0 < n.
      • it follows by definition of divisibility that r0|n.
      • if r0 is prime, then r0 is a prime number that divides n, and we are done. If r0 is not prime, then
      • r0 = r1s1 where r1 and s1 are integers and 1 < r1 < r0
      • etc.
counterexamples and divisibility
Counterexamples and Divisibility
  • Is it true or false that for all integers a and b, if a|b and b|a then a = b?
    • Starting Point: Suppose a and b are integers such that a|b and b|a.
      • b = ka & a =lb (for some integers k and l)
      • b = ka = k(lb) = (kl)b
      • factor b assuming b≠0
      • 1 = kl
      • Thus, k = l = 1 or k = l = -1
unique factorization theorem
Unique Factorization Theorem
  • Theorem 3.3.3 Fundamental Theorem of Arithmetic
    • Given any integer n > 1, there exist a positive integer k, distinct prime numbers p1, p2, …, pk, and positive integers e1, e2, … , ek such that
    • n = p1e1p2e2 p3e3 p4e4 … pkek,
    • and any other expression of n as a product of prime numbers is identical to this except, perhaps, for the order in which the factors are written.
standard factored form
Standard Factored Form
  • Given any integer n > 1, the standard factored form of n is an expression of the form
    • n = p1e1 p2e2 p3e3 p4e4 … pkek,
    • where k is a positive integer; p1, p2, … ,pk are prime numbers; e1, e2, … , ek are positive integers; and p1 < p2 < … < pk.
example
Example
  • Write 3,300 in standard factored form.
    • 3,300 = 100 * 33 = 4 * 25 * 3 * 11
    • = 2 * 2 * 5 * 5 * 3 * 11
    • = 22 * 52 * 3 * 11