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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
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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 • 7.1-7.4 An Introduction to Cryptography • 8.1-8.8 Complex Numbers
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
2.3 Linear Diophantine Equations • Proposition
2.3 Linear Diophantine Equations • Process for Solving
2.3 Linear Diophantine Equations • Process for Solving
2.3 Linear Diophantine Equations • Example • Example
2.3 Linear Diophantine Equations • Example
2.5 Prime Numbers • Definitions
2.5 Prime Numbers • Proposition 2.51 • Euclid Theorem 2.52 • Theorem 2.52 • Unique Factorization Theorem 2.54
2.5 Prime Numbers • Theorem 2.55 • Proposition 2.56
2.5 Prime Numbers • Theorem 2.57 • Theorem 2.58
2.5 Prime Numbers • Example • Example • Example
2.5 Prime Numbers • Example
3.1 Congruence • Definition • Proposition 3.11
3.1 Congruence • Example
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.
3.2 Tests for Divisibility • Example
3.4 Modular Arithmetic • Definitions
3.4 Modular Arithmetic • Fermat’s Little Theorem • Corollary 3.43
3.4 Modular Arithmetic • Example • Example
3.5 Linear Congruences • Definition • Linear Congruence Theorem 3.54
3.5 Linear Congruences • Example • Example • Example • Example
