Lecture 4

Navigation

• Back to Main Page.
• Back to the Lectures.
• Before the Lecture
• Related Problems
• In Class
• Outcomes

Before the Lecture

• Read Section 1.4:
  • Fundamental Theorem of Arithmetic (Theorem 1.51 and its corollaries).
  • Divisibility (Lemma 1.54), GCD and LCM (Proposition 1.55) using prime factorization.
  • $\operatorname{lcm}(a,b) \cdot \gcd(a,b) =\left| a \cdot b \right|$ (Proposition 1.56).
• Watch the videos related to this section (after reading it):
  • Problems 1.68 (iv), (v).
• Read Section 1.5:
  • Congruences (Proposition 1.58 and Corollary 1.59 are important).
  • Application to rule out number which are not perfect squares (Example 1.61).
  • Skip Proposition 1.62 (although it is very nice!).
  • Fermat's Theorem (Theorem 1.64).
  • Divisibility Criteria (like Corollary 1.65).
  • Congruences with powers (Corollary 1.67 and Example 1.68).
  • Solving linear congruences (Theorem 1.69, Examples 1.71 and 1.72).
  • Chinese Remainder Theorem (Theorem 1.73, Examples 1.74 and 1.75, as well as Proposition 1.76 and Example 1.77).
• Watch the videos related to this section (after reading it):
  • General remarks on congruences.
  • Powers in congruences.
  • Corollary 1.65 (divisibility criteria for 3 and 9).
  • Problems 1.77(vi), (vii), 1.78(ii) (and extra example), 1.91(i).
  • CRT with non-relatively prime moduli.
  • Computations with Sage.
• Write down all questions about the above topics to bring to our (online) lecture. (You can also type them in the file "Questions.tex" in SageMathCloud.) Comments about the videos are welcome!
• Work on the assigned problems for these sections. (See Related Problems below.) You don't need to finish them, but try to work on as many as you can and the bring your questions to class.

 

Related Problems

The "turn in" problems are due on 06/12 by 11:59pm.

Section 1.4:	Turn in: 1.68(ii), 1.69(i), 1.76(ii).
	Extra Problems: 1.68, 1.69, 1.70(i), 1.76(ii).

 

Section 1.5:	Turn in: 1.77(viii), 1.78(iii), (v), (vi), 1.91(ii).
	Extra Problems: 1.77, 1.78, 1.79, 1.80, 1.81, 1.83, 1.85, 1.86, 1.91, 1.95.

 

In Class

In class:

• We will discuss the reading and pace.
• I will discuss the main points.
• I will answer questions about the sections covered.
• I will answer any other questions.
• We can work on the HW problems.

 

Outcomes

After the assignment (reading and videos before class) and class, you should:

• know how to factor integers into a product of powers of primes and check divisibility and compute GCDs (Problem 1.69) and LCMs (Problem 1.76(ii)) using these factorizations;
• apply this concept to prove basics properties (Problem 1.70(i));
• understand when numbers are congruent;
• be able to reduce numbers module $m$;
• be able to apply congruence to solve problems (Problems 1.79 and 1.85);
• be able to reduce large powers modulo $m$ (Problem 1.81);
• be able to solve linear congruence equations (Problem 1.78);
• be able to solve systems of congruences (Chinese Remainder Theorem - Problems 1.91 and 1.95).