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Number Theory

This lecture by Peter Burkhardt covers the fundamental concepts of divisibility and modular arithmetic, crucial elements in number theory. You will learn about natural numbers, integers, prime and composite numbers, and the properties of divisibility. The lecture explains the notation for divisibility (a | b), introduces congruences, and demonstrates how to express statements using congruences. Additionally, it includes concepts on division with remainder, modular arithmetic, and Little Fermat’s Theorem.

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Number Theory

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  1. Number Theory Lecture 1 Divisibility and Modular Arithmetic (Congruences) Peter Burkhardt

  2. 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 Peter Burkhardt

  3. 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 Peter Burkhardt

  4. 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 Peter Burkhardt

  5. 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. Peter Burkhardt

  6. 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| Peter Burkhardt

  7. 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. Peter Burkhardt

  8. 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 Peter Burkhardt

  9. Congruence and Division with Remainder Dividing a and b by m yields the same remainder. Peter Burkhardt

  10. Basic Properties of Congruences That is, congruence is an equivalence relation. Peter Burkhardt

  11. Modular Arithmetic ” Demonstration Peter Burkhardt

  12. 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 Peter Burkhardt

  13. Have you understood? How can you write the following statements using congruences? (a, b, r e Z, m e N) • m|a • r is the remainder of a divided by m Using congruences, give a sufficient condition for m|a if and only if m|b Peter Burkhardt

  14. Practice (Handouts) ACTIVITY Peter Burkhardt

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