Number Theory By, B. R. Chandavarkar CSE Dept., NITK Surathkal - PowerPoint PPT Presentation

Basic Concepts in Number Theory By, B. R. Chandavarkar CSE Dept., NITK Surathkal

• Natural number : predecessor and successor. Example: 1, 2, 3, … . • Whole number : natural number with 0. • A factor of number is an exact divisor of that number. – 1 is a factor of every number – Every number is a factor of itself – Every factor of a number is an exact divisor of that number – Every factor is less than or equal to the given number – Number of factors of a given number are finite – Every multiple of a number is greater than or equal to that number – The number of multiples of a given number is infinite – Every number is multiple of itself • Perfect number : A number for which sum of all factors is equal to twice the number is called perfect number. Example, 6 (1, 2, 3 and 6) and 28 (1, 2, 4, 7, 14 and 28) • Prime number : The number other than 1 whose only factors are 1 and number itself are called prime numbers.

• Composite number : Numbers having more than two factors. • Divisibility of numbers : – Divisibility by 2: if a number has any of the digits 0, 2, 4, 6, or 8 in its one place – Divisibility by 3: if the sum of the digits is a multiple of 3 – Divisibility by 4: if the number formed by the last two digits is divisible by 4 – Divisibility by 5: if a number has 0 or 5 in its one position. – Divisibility by 6: if a number is divisible by 2 and 3 both – Divisibility by 8: if the number formed by the last three digits is divisible by 8. – Divisibility by 9: if the sum of the digits of a number is divisible by 9 – Divisibility by 10: if the number has 0 in the ones place • Prime factorization : prime numbers are the factors. Example, 24 (2 X 2 X 2 X 3), 980 (2 X 2 X 5 X 7 X 7) • Co-prime Number: Two numbers having only 1 as a common factor. Example: 4 & 15

• Additional Divisibility Rules – If a number is divisible by another number than it is divisible by each of the factors of that number. Example: 24 divisible by 8 and also by the factors of 8 i.e. 1, 2, 4 and 8. – If a number is divisible by two co-prime numbers than it is divisible by their product also. Example: 80 is divisible by 4 and 5 and also by 4 X 5 = 20. – If two given numbers are divisible by a number, then their sum is also divisible by that number. Example: 16 and 20 are both divisible by 5 and also 16 + 20. – If two given numbers are divisible by a number, then their difference is also divisible by that number. Example: 35 and 20 are both divisible by 5 and also 35 – 20. • Highest Common Factor (HCF) OR Greatest Common Divisor (GCD) : HCF or GCD of two or more given numbers is the highest of their common factors. – Example: (i) 20, 28 and 36 – 4

• Lowest Common Multiple (LCM) : LCF of two or more given numbers is the lowest of their common factors. – Example: (i) 12 and 18 – 36 (ii) 24 and 90 (iii) 40, 48 and 45 (iv) 20, 25 and 30 • Integer : collection of whole numbers and negative numbers. • Fraction : A fraction is a number representing part of a whole. • Improper Fraction : numerator is bigger than denominator. • Rational Number : Number that can be expressed in the form p/q, where p and q are integers and q ≠ 0. • Irrational Number: Number that cannot be expressed in the form p/q, where p and q are integers and q ≠ 0. • Real Number : Collection of rational and irrational numbers.

Important Sets • Definition 1.1 . 1. A set is just a collection of elements. We usually denote a set by enclosing its elements in braces “{ }” . [So, {1, 2, 3, 4} is a set whose elements are the numbers 1, 2, 3, and 4.] • Sets don’t need to have numbers as elements, but they likely will in this course. • Note that the order that we write the elements of the set does not matter, all that matters is the content, i.e., what elements it has. 2. An element of a set is said to belong to the set. We use the symbol “ --- ” for “belongs to” and “ --- ” for “does not belong to”, such as in:

3. We have that denotes the set of natural numbers. [The ellipsis here means “continues in the same way” .] Careful: Some authors exclude zero from the set of natural numbers. We will use instead and refer as the set of positive integers. [Note that zero is neither positive nor negative!] 4. We define the set of integers as 5. We define the set of rational as

• A theorem is a statement [or proposition] whose validity can be deduced from its assumptions by logical steps. So, it is something that you can deduce [the key word here] from other facts. • On the other hand, in mathematics often there is a hierarchy for theorems: – The term Theorem is reserved for statements that have greater importance. You probably know a few: Pythagoras’ Theorem, Fundamental Theorem of Arithmetic, Thale’s Theorem, etc. – When a theorem is useful to us, but is of limited universal importance, the term Proposition is used. It is basically a “minor theorem” . – A Lemma is a theorem whose main purpose is to help prove one or more statements [which can be either full Theorems or mere Propositions]. – Finally, a Corollary is a result, which can be of some relative importance, but is an immediate [or almost immediate] consequence of a previous Theorem or Proposition.

• Note that we must always have a Proposition or Theorem, and never a Corollary, following a Lemma. In the same way, a Corollary always comes after a Proposition or a Theorem, but never after a Lemma.

• We now review the concepts of greatest common divisor, which we shall abbreviate by GCD, and least common multiple, which we shall abbreviate by LCM. • The names already tell us what they mean: the GCD of two integers a and b is the largest integer that divides a and b [at the same time], and the LCM is the smallest positive integer that is a multiple of a and of b [at the same time]. • We shall denote them gcd(a, b) and lcm(a, b) respectively. • Note that for all positive integers a and b, we have that gcd(a, b) ≥ 1 and lcm(a, b) ≤ ab. • Moreover, since a divisor of a number is always less than or equal to the number itself, and a multiple of a number is always greater than or equal to the number itself, we can also conclude that gcd(a, b) ≤ min(a, b) [where min(a, b) is the minimum between a and b] and lcm(a, b) ≥ max(a, b) [where max(a, b) is the maximum between a and b]. • In summary: 1 ≤ gcd(a, b) ≤ min(a, b) and max(a, b) ≤ lcm(a, b) ≤ ab.

• GCD using Euclidean Algorithm • LCM

Lemma: The product of two or more integers of the form 4n+1 is of the same form.

Congruence

Each residue class modulo n may be represented by any one of its members, although we usually represent each residue class by the smallest nonnegative integer which belongs to that class (since this is the proper remainder which results from division). Any two members of different residue classes modulo n are incongruent modulo n. Furthermore, every integer belongs to one and only one residue class modulo n.

The set of integers {0, 1, 2, … , n − 1} is called the least residue system modulo n . Any set of n integers, no two of which are congruent modulo n, is called a complete residue system modulo n. It is clear that the least residue system is a complete residue system , and that a complete residue system is simply a set containing precisely one representative of each residue class modulo n. The least residue system modulo 4 is {0, 1, 2, 3}. Some other complete residue systems modulo 4 are: {1, 2, 3, 4}, {13, 14, 15, 16}, {− 2, − 1, 0, 1}, {− 13, 4, 17, 18}, {− 5, 0, 6, 21}, {27, 32, 37, 42} Some sets which are not complete residue systems modulo 4 are: {− 5, 0, 6, 22} since 6 is congruent to 22 modulo 4. {5, 15} since a complete residue system modulo 4 must have exactly 4 incongruent residue classes.

Show that if a, b, and c are integers, then [a, b ]| c if and only if a | c and b | c .

Find the least positive residue of 2 644 mod 645

Find all solutions of 9x is congruent to 12 (mod 15) Ans: 8, 13, and 18(3) Find all solutions of 7x congruent to 4 (12) Ans: -20 (4)

GALOIS FIELD – GROUP • Group/ Albelian Group: A group G or {G, .} is a set of elements with a binary operation denoted by . , that associates to each ordered pair (a, b) of elements in G an element (a . b) such that the following properties are obeyed: – Closure: If a & b belong to G, then a . b also belongs to G. – Associative: For elements a, b & c in G, a . (b . c) = (a . b) . c. – Identity element: There is an element e in G such that a . e = e . a = a, for all a in G. – Inverse element: For each element a in G there is an element a’ in G such that a . a’ = a’ . a= e. – Commutative: for all elements a & b in G, a . b = b . a.