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# Math 135 Midterm Exam-AID Session

Math 135 Midterm Exam-AID Session. Agenda. 2.3 Linear Diophantine Equations 2.5 Prime Numbers 3.1 Congruence 3.2 Tests for Divisibility 3.4 Modular Arithmetic 3.5 Linear Congruences. Agenda. 3.6 The Chinese Remainder Theorem 3.7 Euler Fermat Theorem

## Math 135 Midterm Exam-AID Session

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1. Math 135 Midterm Exam-AID Session

2. Agenda • 2.3 Linear Diophantine Equations • 2.5 Prime Numbers • 3.1 Congruence • 3.2 Tests for Divisibility • 3.4 Modular Arithmetic • 3.5 Linear Congruences

3. Agenda • 3.6 The Chinese Remainder Theorem • 3.7 Euler Fermat Theorem • 7.1-7.4 An Introduction to Cryptography • 8.1-8.8 Complex Numbers

4. 2.3 Linear Diophantine Equations • Definition • Linear Diophantine Equation is an equation in one or more unknowns with integer coefficients, for which integer solutions are sought • Linear Diophantine Equation Theorem

5. 2.3 Linear Diophantine Equations • Proposition

6. 2.3 Linear Diophantine Equations • Process for Solving

7. 2.3 Linear Diophantine Equations • Process for Solving

8. 2.3 Linear Diophantine Equations • Example • Example

9. 2.3 Linear Diophantine Equations • Example

10. 2.5 Prime Numbers • Definitions

11. 2.5 Prime Numbers • Proposition 2.51 • Euclid Theorem 2.52 • Theorem 2.52 • Unique Factorization Theorem 2.54

12. 2.5 Prime Numbers • Theorem 2.55 • Proposition 2.56

13. 2.5 Prime Numbers • Theorem 2.57 • Theorem 2.58

14. 2.5 Prime Numbers • Example • Example • Example

15. 2.5 Prime Numbers • Example

16. 3.1 Congruence • Definition • Proposition 3.11

17. 3.1 Congruence • Example

18. 3.2 Tests for Divisibility • Theorem 3.21 • A number is divisible by 9 if and only if the sum of its digits is divisible by 9. • Theorem 3.22 • A number is divisible by 3 if and only if the sum of its digits is divisible by 3. • Proposition 3.23 • A number is divisible by 11 if and only if the alternating sum of its digits is divisible by 11.

19. 3.2 Tests for Divisibility • Example

20. 3.4 Modular Arithmetic • Definitions

21. 3.4 Modular Arithmetic • Fermat’s Little Theorem • Corollary 3.43

22. 3.4 Modular Arithmetic • Example • Example

23. 3.5 Linear Congruences • Definition • Linear Congruence Theorem 3.54

24. 3.5 Linear Congruences • Example • Example • Example • Example

25. 3.6 The Chinese Remainder Theorem

26. 3.6 The Chinese Remainder Theorem

27. 3.6 The Chinese Remainder Theorem

28. 3.6 The Chinese Remainder Theorem

29. 3.7 Euler-Fermat Theorem

30. 7.1 Cryptography

31. 7.2 Private Key Cryptography

32. 7.3 Public Key Cryptography

33. 7.4 RSA Scheme

34. 7.4 RSA Scheme

35. 7.4 RSA Scheme

36. 7.4 RSA Scheme

37. 7.4 RSA Scheme

38. 7.4 RSA Scheme

39. 8.1 Quadratic Equation

40. 8.2 Complex Numbers

41. 8.2 Complex Numbers

42. 8.3 Complex Plane

43. 8.3 Complex Plane

44. 8.4 Properties of Complex Numbers

45. 8.4 Properties of Complex Numbers

46. 8.4 Properties of Complex Numbers

47. 8.4 Properties of Complex Numbers

48. 8.5 Polar Representation

49. 8.5 Polar Representation

50. 8.5 Polar Representation

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