Lesson 9: Primes and Division

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Lesson 9 Primes and Division

• Objectives
• State basic properties of divisibility
• State the fundamental theorem of algebra
• Use the sieve of Eratosthenes method to find all
  of the primes of a number not exceeding n
• Prove that there are infinitely many primes
• Find the GCD and LCM of numbers

• Outline
• Division
• Primes Factoring
• Properties of Primes
• GCD, LCM
• Reading Section 2.4
• HW due 2/16
• 2.4 9b-d, 12b-e, 13, 18bc, 24b, 28 b-d, 30b-d,
  36

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Ackermans Function

• http//www.kosara.net/thoughts/ackermann.html

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Divisibility

• a divides b if there is an integer c such that
• b ac
• a is a factor of b
• b is a multiple of a (the cth multiple of a)
• a b
• If a does not divide b, we say
• a ?b
• a b if ?c(acb)
• universe of discourse integers

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Divisibility

• 4 15
• 4 16
• 16 4
• 3 0
• 13 13

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In Class

• Write down as many properties of divisibility as
  you can think of.
• (dont look ahead in the notes!!)

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Divisibility Properties

• if a b and a c, then a (b c)
• if a b, then a bc for all integers c
• if a b and b c, then a c

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Divisibility Properties

• If a, b, c are integers such that a b and a
  c, then
• a mb nc (m, n ? Z)

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Prime Numbers

• A positive number greater than 1 is prime if and
  only if the only positive factors of p are 1 and
  p
• non-primes are called composite
• How to test if a number is prime?

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Fundamental Theorem of Arithmetic

• Every positive integer greater than 1 can be
  written uniquely as a prime or as the product of
  two or more primes where the prime factors are
  written in order of nondecreasing size.
• 64 222222
• 33 3 11
• 175 5 5 7
• 4951 4951

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Prime Factorization

• 462
• 135
• 735
• 768

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Prime Factorization

• If n is a composite integer, then n has a prime
  divisor less than or equal to
• Prove that 379 is prime

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The Sieve of Eratosthenes

• The composites less than 121 must have prime
  factors less than or equal to 11

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Infinitude of Primes

• There are infinitely many primes

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Mersenne Primes

• Mersenne primes are of the form 2p-1, where p is
  prime
• The largest known primes are Mersenne Primes
• Lucas-Lehmer test
• Great International Mersenne Prime Search (GIMPS)
• 2(13466917)-1 (over 4 million digits)
• http//www.isthe.com/chongo/tech/math/prime/mersen
  ne.htmlM13466917

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Distribution of Primes

• The number of primes not exceeding x approaches
  ln(x) as x approaches infinity
• If you randomly choose a number from 1, 106,
  what is the probability that the number will be
  prime?

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Division

• Divisor
• Dividend
• Quotient
• Remainder
• q a/d
• q a div d
• r a mod d

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Greatest Common Divisor

• If a,b are integers, gcd(a,b) is the largest
  integer d such that d a and d b.
• d is called the greatest common divisor of b and
  a
• In-Class
• gcd(63,27)
• gcd(49, 32)
• gcd(14, 18)
• If gcd(a,b) 1, then a,b are relatively prime
• Groups of numbers can be pairwise relatively
  prime

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Least Common Multiple

• lcm(a,b) is the smallest positive integer that is
  divisible by both a and b
• lcm(40, 50)
• lcm(133, 233)
• lcm(5, 50)

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GCD and LCM using Prime Factorization

• GCD of (a,b) p1min(a1,b1) p2min(a2,b2)
  p3min(a3,b3)
• GCD (140, 400)
• LCM of (a,b) p1max(a1,b1) p2max(a2,b2)
  p3max(a3,b3)
• LCM (1000, 625)

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Congruences

• a is congruent to b (modulo m) if m (a-b)
• a ? b (mod m)
• 14 ? 2 (mod 3)
• 27 ? 7 (mod 5)
• 25 ? 2 (mod 6)

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Congruences

• 26 16 (mod 3)
• 15 13 (mod 2)
• 100 50 (mod 25)
• a ? b (mod m) if and only if (a mod m) (b mod
  m)
• a, b are congruent modulo m if and only if there
  is an integer k such that (a b km)

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Hashing

• Hashing functions map many-to-few
• h(k) k mod m
• (where k is number of keys and m is number of
  slots)

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Pseudorandom Number generation

• modulus m
• multiplier a (greater or equal to 2, less than
  m)
• increment c (greater or equal to 0, less than m)
• seed x0 (greater or equal to 0, less than m)
• xn1 (axn c) mod m
• Find sequence for m 7, a 5, c 1, x0 2