Prep Math Competition, Lec. 1Peter Burkhardt1 Number Theory Lecture 1 Divisibility and Modular Arithmetic (Congruences)

Prep Math Competition, Lec. 1Peter Burkhardt2 Basic Definitions and Notations (1)  N = {1,2,3,…} denotes the set of natural numbers  Z = {…,-3,-2,-1,0,1,2,3,…} denotes the set of integers

Prep Math Competition, Lec. 1Peter Burkhardt3 Basic Definitions and Notations (2) Divisibility (1) Let a, b e Z, a not equal to zero. We say a divides b if there exists an integer k such that

Prep Math Competition, Lec. 1Peter Burkhardt4 Basic Definitions and Notations (3) Divisibility (2) In this case we write a|b Sometimes we say that:  b is divisible by a, or  a is a factor of b, or  b is a multiple of a

Prep Math Competition, Lec. 1Peter Burkhardt5 Basic Definitions and Notations (4) Prime and Composite Numbers A natural number p > 1 is called a prime number, or, simply, prime, if it is divisible only by itself and by 1. P = {2,3,5,7,…} denotes the set of prime numbers. Otherwise the number is called composite.

Prep Math Competition, Lec. 1Peter Burkhardt6 Properties of Divisibility  a|b ” a|bc for each integer c  a|b and b|c ” a|c  a|b and a|c ” a|(bx + cy) for any x, y e Z  a|b and a, b not equal to zero ” |a| £ |b|

Prep Math Competition, Lec. 1Peter Burkhardt7 Division with Remainder Let m,a e Z, m > 1. Then, there exist uniquely determined numbers q and r such that a = qm + r with 0 £ r < m Obviously, m|a if and only if r = 0.

Prep Math Competition, Lec. 1Peter Burkhardt8 Congruences Let a, b e Z, m e N. We say a is congruent to b modulo m if m|(a-b) and we write

Prep Math Competition, Lec. 1Peter Burkhardt9 Congruence and Division with Remainder Dividing a and b by m yields the same remainder.

Prep Math Competition, Lec. 1Peter Burkhardt10 Basic Properties of Congruences That is, congruence is an equivalence relation.

Prep Math Competition, Lec. 1Peter Burkhardt11 Modular Arithmetic ” Demonstration Demonstration

Prep Math Competition, Lec. 1Peter Burkhardt12 Little Fermat’s Theorem Let a e Z, and p prime. If p does not divide a, then For all a e Z we have

Prep Math Competition, Lec. 1Peter Burkhardt13 Have you understood? How can you write the following statements using congruences? (a, b, r e Z, m e N) 1.m|a 2.r is the remainder of a divided by m Using congruences, give a sufficient condition for m|a if and only if m|b

Prep Math Competition, Lec. 1Peter Burkhardt14 Practice (Handouts)