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Integers and Division Integers and Algorithms

Learning Objectives

- Integers and division.
- Integers and algorithms.

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

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)

Integers and Division

- Theorem if n is a composite integer, then n has

a prime divisor less than or equal to ?n. - demonstration 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.

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

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

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). - Example 43 ? 27 (8) (same mod 3).

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. 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

Integers and Algorithms

- Euclidean algorithm Euclid 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

Integers and Algorithms

- Algorithm procedure gcd(a,b positive

integers) x a y b while y gt 0 begin r x

mod y x y y r end gcd(a,b) is x

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

Integers and Algorithms

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

Integers and Algorithms

- Algorithm procedure base b expansion(npositive

integer) q n k 0 while q gt 0 begin ak

q mod b q ?q/b ? k k 1 end the

base b expansion of n is (ak a0)b

Integers and Algorithms

- Integer operations algorithms to perform

operations directly on binary expansions. - Example addition O(n)
- procedure add(a,bpositive integers) c 0 for

j 0 to n-1 begin d ?(aj bj c)/2 ?

sj aj bj c - 2d c d end sn

c the binary expansion of the sum is

(snsn-1 s1s0)2