The Integers and Division

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

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 | (b+c); –if a | b, then a | bc; –if a | b and b | c, then a | c.

Exercise 2.3a

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 = 2 2  5 2, 641 = 641, 999 = 3 3  37, and 1024 = 2 10.

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 = –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 = 7 2  11  13

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 < 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  How about: 101 = 11  ? -11 = 3(-4) + 1. How about: -11 = 3(-3) - 2?

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.

Greatest Common Divisors -Cont Procedure to find gcd(a, b): –Find the prime factorization of a and b. –If a = p 1 a 1 p 2 a 2 … p n a n, b = p 1 b 1 p 2 b 2 … p n b n, then gcd(a, b) = p 1 min(a 1,b 1 ) p 2 min(a 2,b 2 ) … p n min(a n,b n ) –Examples 120 = 2 3  3  5 and 500 = 2 2  5 3 = 2 2  3 0  5 3 gcd(120, 500) = 2 2  3 0  5 1 = 20.

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 a 1, a 2, …, a n are pairwise relatively prime if gcd(a i, a j )=1 whenever 1  i<j  n. –Examples 10, 17 and 21 are pairwise relatively prime. 10, 17 and 24 are not pairwise relatively prime.

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 = p 1 a 1 p 2 a 2 … p n a n, b = p 1 b 1 p 2 b 2 … p n b n, then lcm(a, b) = p 1 max(a 1,b 1 ) p 2 max(a 2,b 2 ) … p n max(a n,b n ) –Example lcm( , ) = Theorem  a  +  b  + ab = gcd(a, b)  lcm(a, b)

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 < m and a = qm + r. –Examples 17 mod 5 = 2 (since 17 = 3  ) -133 mod 9 = mod 101 = 82 –The function f m : Z → {0, 1, 2, …, m-1}, where f m (a) = a mod m is onto, but not one-to-one.

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 = ¬  14 (mod 6) I used above notation for lack of strike  in PP.

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: a+c  b+d (mod m), and ac  bd (mod m).

Applications of Congruence Hashing Functions Pseudorandom Numbers –Linear congruential method Cryptology –Caesar cipher

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 m=111, the record with k= is assigned to location 14 since h( ) = mod 111 = 14. Collision may occur since h(k) is not one-to-one. –Resolve by assigning next free location.

Pseudorandom Numbers Linear congruential method –Choose: modulus m, multiplier a, increment c, and seed x 0, with 2  a < m, 0  c, x 0 < m –Generate the sequence {x n } x n+1 = (a x n + c) mod m. –Example m = 9, a = 7, c = 4, and x 0 = 3: x 1 = 7x 0 +4 mod 9 = 7  mod 9 = 25 mod 9 = 7 x 2 =8, x 3 =6, x 4 =1, x 5 =2, x 6 =0, x 7 =4, x 8 =5, x 9 =3. –Usually, a pure multiplicative generator is used: Increment c=0, modulus m=2 31 – 1, multiplier a=7 5 =16,807.

Cryptology Caesar’s 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) = (12+3) 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.

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  ) mod 26 = 73 mod 26 = 21  V. –K is replaced by V in the encrypted message.