CHAPTER 2 NUMBER THEORY NUMBER SYSTEM COMPUTER ARITHMETIC

CHAPTER 2 NUMBER THEORY, NUMBER SYSTEM & COMPUTER ARITHMETIC, CRYPTOGRAPHY BY MISS FARAH ADIBAH ADNAN IMK

CHAPTER OUTLINE - I • 2. 1 INTRODUCTION TO NUMBER THEORY 2. 1. 1 DIVISIBILITY • PRIME NUMBERS • FACTORIZATION 2. 1. 2 GREATEST COMMON DIVISOR (GCD) • EUCLIDEAN ALGORITHM 2. 1. 3 MODULAR ARITHMETIC • 2. 2 NUMBER SYSTEM & COMPUTER ARITHMETIC • 2. 3 CRYPTOGRAPHY

2. 1 INTRODUCTION TO NUMBER THEORY • The theory of number is part of discrete mathematics and involving the integers and their properties. • The concept of division of integers is fundamental to computer arithmetic. • This chapter involves algorithms – used to solves many problems; searching a list, sorting finding the shortest path, find greatest common divisor, etc.

2. 1. 1 DIVISIBILITY • Number theory is concerned with the properties of integers. • One of the most important is divisibility. Definition 2. 1 Let and be integers with We say that divides if there is an integer such that This is denoted by Another way to express this is that is a multiple of .

EXAMPLE 2. 1 Determine whether they are divisible or not. a) . b) . c) .

Basic Properties of Divisibility Let represent integers. Then i. . ii. . iii. .

2. 1. 1. 1 PRIME NUMBERS Definition 2. 2 A number that is divisible only by 1 and itself is called prime number. An integer that is not prime is called composite, which means that must expressible as a product of integers with

EXAMPLE 2. 2 Determine each number is prime number or not. a) 7 b) 9

The first 1000 primes are listed below. 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 193 197 199 211 223 227 229 233 239 241 257 263 269 271 277 281 283 293 307 311 313 317 331 337 349 353 359 367 373 379 383 389 397 401 409 419 421 433 439 443 449 457 461 463 467 479 487 491 499 503 509 521 523 541 547 557 563 569 571 577 587 593 599 601 607 613 617 619 631 643 647 653 659 661 673 677 683 691 709 719 727 733 739 743 751 757 761 769 773 787 797 809 811 823 827 829 839 853 857 859 863 877 881 883 887 907 911 919 929 937 941 947 953 967 971 977 983 991 997

2. 1. 1. 2 FACTORIZATION • Theorem 2. 1: The Fundamental Theorem of Arithmetic • Any integer is a product of primes uniquely, up to the order of primes. It can be factored in a unique way as: where are prime numbers and

EXAMPLE 2. 3 Find prime factorizations of: a) 91 b) 11011

2. 1. 2 GREATEST COMMON DIVISOR (GCD) Definition 2. 3 • The greatest common divisor of and is the largest positive integer dividing both and is denoted by either . When , we say and are relatively primes.

Steps to Find the Gcd • If you can factor and into primes, do so. For each prime number, look at the powers that it appears in the factorizations of and . Take the smaller of the two. Put these prime powers together to get the gcd. This is easiest understand by examples: gcd (576, 135) =

Steps to Find the Gcd ii) gcd Note that if prime does not appear in factorization, then it cannot appear in the gcd. • Suppose and are large numbers, so it might not be easy to factor them. The gcd can be calculated by a procedure known as the Euclidean algorithm. It goes back to what everyone learned in grade school: division with remainder.

2. 1 THE EUCLIDEAN ALGORITHM • Suppose that is greater than . If not, switch and . The first step is divide the larger of the two integers by the smaller; let is larger than , hence represent in the form Where is called the dividend, is called the divisor, is called the quotient, and is called the remainder.

2. 1 THE EUCLIDEAN ALGORITHM • If then continue by representing in the form • Continue this way until the remainder that is zero.

• Let and • The following sequence of steps: • Hence, the greatest common divisor is the last nonzero remainder in the sequence of divisions

• There are two important aspects to this algorithm: v It does not require factorization of the numbers. v It is fast.

EXAMPLE 2. 4 Compute

2. 1. 3 MODULAR ARITHMETIC • Sometimes we care about the remainder of an integer when it is divided by some specified positive integer. Definition 2. 4 Let be a fixed positive integer. For any integer , is the remainder upon dividing by . Eg:

2. 1. 3 MODULAR ARITHMETIC • One of the most basic and useful in number theory is modular arithmetic, or known as congruence. Definition 2. 5 If and are integers and is a positive integer, then and are said to be congruent modulo if . This is written .