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

2. 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

3. 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

4. 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

5. 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.

6. 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|

7. 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.

8. 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

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

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

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

12. 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

13. 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

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