Integers and Division Integers and Algorithms

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Integers and DivisionIntegers and Algorithms
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Learning Objectives

• Definition of integer division.
• Properties of division.
• Prime numbers and some of their properties.
• The division algorithm.
• Greatest Common Divisor and Least Common
  Multiple.
• Congruences, modular arithmetic.

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Learning Objectives

• Integers and algorithms
• the Euclidian algorithm
• the extended Euclidian algorithm
• Further results in number theory.
• Some applications of number theory
• computer arithmetic with large numbers
• public key cryptography

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Integers and Division

• Division if a and b are integers with a ? 0, we
  say that a divides b if there is an integer c
  such that b ac.
• Factor when a divides b, we say that a is a
  factor of b.
• Multiple when a divides b, we say that b is a
  multiple of a.
• Notation a b (a divides b).
• Example 3 27 but not 2 17

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Integers and Division

• Prime a positive integer p greater than 1 is
  called prime if the only positive factors of p
  are 1 and p.
• Composite a positive integer that is greater
  than 1 and is not prime.
• List of primes 2,3,5,7,11,13,17,19,23,29,31,37,41
  ,43,47,53
• The fundamental theorem of arithmetic every
  positive integer can be written as the product of
  primes, where the prime factors are written in
  order of increasing size.
• Examples of prime factorizations (Maple
  ifactor(n)) gt ifactor(31722722304)

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Integers and Division

• Theorem if n is a composite integer, then n has
  a prime divisor less than or equal to ?n.
• proof there exist a and b, n a.b. We need to
  have a ? ?n or b ? ?n because otherwise a.b
  would be greater than n. Let us suppose that a ?
  ?n. If a is prime, QED. If it is a composite,
  then a still has a prime divisor smaller than ?n,
  QED.
• Example find the prime factorization of 97.
• Try to divide 97 by increasingly great prime
  numbers, stopping at ? 979. 2,3,5,7 do not
  divide 97, so it is prime.

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Integers and Division

• Division algorithm let a be an integer and d a
  positive integer. Then there are unique integers
  q and r, with 0 ? r lt d, such that a dq r.
• Divisor d.
• Dividend a.
• Quotient q.
• Remainder r. Also called modulus.
• Examples irem(145,34) 9
• iquo(145,34) 4

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Integers and Division

• Greatest Common Divisor let a and b be two
  integers, not both zero. The largest integer d
  such that d a and d b is called the greatest
  common divisor of a and b, and is noted gcd(a,b).
  (take all common - divisors with smallest
  exponent).
• Least Common Multiple the least common multiple
  of the positive integers a and b is the smallest
  positive integer that is divisible by both a and
  b, and it is noted lcm(a,b). (take all divisors
  with largest exponent).
• Examples
• igcd(129, 39) 3
• ilcm(129,39) 1677

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Integers and Division

• Properties
• ab gcd(a,b) . lcm(a,b)
• Modular arithmetic
• two integers a and b are congruent modulo m (m
  integer gt 0) if m divides a-b. It is denoted as
  a ? b (mod m).
• We can also say that they have the same remainder
  when divided by m.
• Example 43 ? 27 (8) (same modulus 3).

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Integers and Division

• Properties of congruences
• If a ? b (mod m) and c ? d (mod m), thena c
  ? b d (mod m)
• Proof If a ? b (mod m), then there is a k such
  as a b km. There is also an l such as c b
  lm.By additive property of equality, we have
  ac bd(kl)m, and a c ? b d (mod m).
  QED

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Integers and Division

• Applications
• Hashing functions permits to allocate a memory
  space to a piece of data. For example, if storage
  available is a table of size 20, a number n to
  store will be associated the cell n mod 20.
  Problem potential collisions.
• Cryptology science of secret messages.Julius
  Caesar encryption code shift each letter 3
  positions forward in the alphabet (a --gt d, z--gt
  c, ).f(p) (p3) mod 26, where p is a code for
  each letter between 0 and 25 (A ? 0, , Z?
  25).AN EXAMPLE --gt 0 13 4 23 0 12 15 11
  4--gt 3 16 7 0 3 15 18 14 7 --gt DQ HADPSOH

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Integers and Algorithms

• Euclidean algorithmEuclid proposed this
  algorithm to calculate the gcd of two
  numbers.Let abqr, where a,b,q,r are integers.
  Then gcd(a,b)gcd(b,r).If d divides a and b, it
  divides also r because ra-bq. So all common
  divisor of a and b is also a common divisor of b
  and r.If d divides b and r, it divides also a
  because abqr. So all common divisors of b and r
  are also common divisors of a and b.
• Examplegcd(662,414) gcd(414,248)
  gcd(248,166) gcd(166,82) gcd(82,2) 2

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Integers and Algorithms

• Algorithm procedure gcd(a,b positive
  integers)x ay bwhile y gt 0begin r x
  mod y x y y rend gcd(a,b) is x

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Integers and Algorithms

• Base b expansion representation of n such that
• where n is a positive integer, k is
  a nonnegative integer, a0, a1, , ak
  are nonnegative integers lt b
• ak ? 0
• Example binary expansion, where all digits are
  either 0 or 1. Hexadecimal expansion, where all
  digits are 0,1, , 9, A, B, C, D, E, F

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Integers and Algorithms

• Examples
• (101011)2 25 23 21 20 43
• 145 (10010001) 2
• (3A5)16 3.162 10.161 5
• 145 (91)16

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Integers and Algorithms

• Algorithmprocedure base b expansion(npositive
  integer)q nk 0while q gt 0begin ak
  q mod b q ?q/b ? k k 1end the
  base b expansion of n is (aka0)b

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Integers and Algorithms

• Integer operations algorithms to perform
  operations directly on binary expansions.
• Example addition O(n)
• procedure add(a,bpositive integers)c 0for
  j 0 to n-1begin d ?(aj bj c)/2 ?
  sj aj bj c - 2d d is the next
  carry c dendsn c the binary
  expansion of the sum is (snsn-1 s1s0)2