The Integers and Division

1
The Integers and Division
2
Outline

• Division Factors, multiples
• Exercise 2.3
• Primes The Fundamental Theorem of Arithmetic.
• The Division Algorithm
• Greatest Common Divisors Relatively prime
• Least Common Multiples
• Modular Arithmetic Congruence
• Applications of Congruence Cryptology

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Division

• Definition Let a and b be integers with a?0.
• Then, we say that a divides b (and we note a
  b) if there is an integer c such that b ac.
• a is called a factor of b, and b is multiple of
  a.
• We note a b when a does not divide b
• I used above notation for lack of strike vertical
  in PP.
• Examples 3 12, but 3 14
• Note P(a, b) a b is a predicate, with values
  True or False.
• Theorem Let a, b, c be integers with a ? 0. Then,
• if a b and a c, then a (bc)
• if a b, then a bc
• if a b and b c, then a c.

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Exercise 2.3a
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Primes

• Definition A positive integer p greater than 1
  is called prime if the only positive factors of p
  are 1 and p.
• A positive integer that is greater than 1 and is
  not prime is called composite.
• Examples 7 is prime. 9 is composite.
• Note 1 is not prime, nor composite.
• Some primes 2, 3, 5, 7, 11, 13, 17, 19, 23, 29,
  31, 37, 41, 43, 47
• The Fundamental Theorem of Arithmetic
• Every positive integer can be written uniquely
  as the product of primes, in increasing order.
• Examples. 100 22 ? 52, 641 641, 999 33 ?
  37, and 1024 210.

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Primes Cont.

• Theorem If n is a composite integer, then n has
  a prime divisor less than or equal to ?n.
• An integer n is prime if it is not divisible by
  any prime less than or equal to ?n.
• 101 is prime, since 101 is not divisible by 2, 3,
  5, or 7 (the only primes less or equal than
  ?101.)
• Prime factorization of 7007
• Divide 7007 by primes, starting with 2, 3, 7
  7007/7 1001.
• Divide 1001 by primes, starting with 7 1001/7
  143.
• Divide 143 by primes, starting with 7 143/11
  13.
• Stop, since 13 is prime. 7007 72 ? 11 ? 13

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The Division Algorithm

• The 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.
• d is called the divisor,
• a is called the dividend,
• q is called the quotient,
• r is called the reminder.
• Examples 101 11 ? 9 2. How about 101 11 ?
  8 13?
• -11 3(-4) 1. How about -11 3(-3) -
  2?

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Greatest Common Divisors

• Definition Let a and b be integers, not both
  zero. The largest integer d such that d a and d
  b , denoted by gcd(a, b), is called the
  greatest common divisor of a and b.
• Examples gcd(24, 36) 12.
• gcd(17, 22) 1.

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Greatest Common Divisors -Cont

• Procedure to find gcd(a, b)
• Find the prime factorization of a and b.
• If a p1a1 p2a2 pnan, b p1b1 p2b2 pnbn,
  then
• gcd(a, b) p1min(a1,b1) p2min(a2,b2)
  pnmin(an,bn)
• Examples
• 120 23 ? 3 ? 5 and 500 22 ? 53 22 ? 30 ?
  53
• gcd(120, 500) 22 ? 30 ? 51 20.

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Relatively Prime Integers

• Definition The integers a and b are relatively
  prime if gcd(a, b) 1.
• Example 17 and 22 are relatively prime.
• Definition The integers a1, a2, , an are
  pairwise relatively prime if gcd(ai, aj)1
  whenever 1?iltj?n.
• Examples
• 10, 17 and 21 are pairwise relatively prime.
• 10, 17 and 24 are not pairwise relatively prime.

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Least Common Multiples

• Definition Let a and b be positive integers. The
  least common multiple of a and b is the smallest
  positive integer that is divisible by both a and
  b.
• It is denoted by lcm(a, b).
• If a p1a1 p2a2 pnan, b p1b1 p2b2 pnbn,
  then
• lcm(a, b) p1max(a1,b1) p2max(a2,b2)
  pnmax(an,bn)
• Example lcm(233572, 2433) 243572.
• Theorem ?a?? ?b??
• ab gcd(a, b) ? lcm(a, b)

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Modular Arithmetic

• Definition Let a be an integer and m a positive
  integer. a mod m denotes the reminder when a is
  divided by m.
• a mod m r, where 0 ? r lt m and a qm r.
• Examples
• 17 mod 5 2 (since 17 3 ? 5 2.)
• -133 mod 9 2
• 2001 mod 101 82
• The function fm Z ? 0, 1, 2, , m-1, where
  fm(a) a mod m is onto, but not one-to-one.

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Congruence

• Definition If a and b are integers and m a
  positive integer, then a is congruent to b modulo
  m (a ? b (mod m)) if m divides (a b).
• Note a ? b (mod m) ? a mod m b mod m
• Examples
• 17 ? 5 (mod 6), since 17-5 12 6 ? 2 is a
  multiple of 6.
• Note also that 17 mod 6 5 mod 6 5.
• 24 ? 14 (mod 6)
• I used above notation for lack of strike ? in PP.

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Congruence Cont.

• Theorem ?m?Z ?a?Z ?b?Z
• a?b (mod m) ? ?k?Z a b km
• Theorem
• If a?b (mod m) and c?d (mod m), then
• ac ? bd (mod m), and
• ac ? bd (mod m).

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Applications of Congruence

• Hashing Functions
• Pseudorandom Numbers
• Linear congruential method
• Cryptology
• Caesar cipher

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Hashing Functions

• Records are identified by a key (integer k).
• For example, using Social Security number
• To record k, assign memory location
• h(k) k mod m, where m is the number of
  available memory locations.
• h(k) is easily evaluated it is also onto.
• Example. If m111, the record with k064212848 is
  assigned to location 14 since h(064212848)
  064212848 mod 111 14.
• Collision may occur since h(k) is not one-to-one.
• Resolve by assigning next free location.

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Pseudorandom Numbers

• Linear congruential method
• Choose modulus m, multiplier a, increment c, and
  seed x0, with 2 ? a lt m, 0 ? c, x0 lt m
• Generate the sequence xn
• xn1 (a xn c) mod m.
• Example m 9, a 7, c 4, and x0 3
• x1 7x04 mod 9 7 ? 3 4 mod 9 25 mod 9 7
• x28, x36, x41, x52, x60, x74, x85, x93.
• Usually, a pure multiplicative generator is used
• Increment c0, modulus m231 1, multiplier
  a7516,807.

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Cryptology

• Caesars encryption process
• Represent each letter by an integer from 0 to 25
• Replace a letter represented by p by the letter
  represented by f(p) (p 3) mod 26.
• Example
• M ? 12, f (12) (123) mod 26 15 ? P
• Meet you in the park is replaced by Phhw brx
  lq wkh sdun
• Decryption. To recover the original message, use
  the inverse function f -1(p) (p - 3) mod 26.

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Cryptology Cont.

• Caesar cipher can be generalized
• Shift cipher
• f(p) (p k) mod 26.
• Affine transformation
• f(p) (ap b) mod 26, where a and be are
  integers chosen so that f is a bijection.
• Example f(p) (7p 3) mod 26, K?
• K ? 10, f (10) (7 ? 10 3) mod 26 73 mod 26
  21 ? V.
• K is replaced by V in the encrypted message.