Division Algorithm and common divisors (1.25-1.34)

• Well-Ordering Axiom statement.
  • This is not true for the integers, think of examples. Then restate it a bit to make it true for the integers.
  • This is not true for the real numbers, think of examples.
• Example Theorem. Make sure you understand how it works. What is the significance of 7?
• Theorem: The division algorithm. Make sure you clearly understand the statement.
• Example 1.25. Construct other similar examples.
• Theorem 1.26. The example theorem should help you.
• Theorem 1.27.
• Theorem 1.28. What does this theorem tell us about the equivalence classes of congruence?
• Definition: common divisor.
• Question 1.29.
• Question 1.30.
• Definition: greatest common divisor.
  • Provide Examples.
• Definition: relatively prime numbers.
• Exercise 1.31.
• Theorem 1.32.
  • Provide Examples.
  • Can you think of any kind of converse?
• Theorem 1.33. Use the previous theorem.
• Theorem 1.34.