Class Summary - 10/09/16

Instructors:	Prof. Isa Hafalir, Mr. Rameel Rizvi
Attendance:	Ayyan, Omar, Areej, AbdulRafie
Homework:	Read: main concepts from Chapter 3.1, 3.2, 3.3, 3.4 (look over the important results); Problems to solve:  3.1.8, 3.2.2, 3.2.7, 3.3.1, 3.3.5 (all parts), 3.4.1 (all parts), 3.4.3, 3.4.8 (all parts)
Challenge Problem:	Let P be the product of the first 100 positive odd integers. Find the largest integer k such that P is divisible by 3^k. (Source: AIME II 2006)
Class Activity:	Covered chapters 3.1-3.4
Concepts:	Defined the concept of multiples: an integer a is a multiple of an integer b if and only if a = mb for some integer m. This mirrors the definition of divisibility: a is divisible by b if and only if a = mb for some integer m.  We briefly covered the use of set notation in generalizing numbers with certain properties, such as when we said m is an element of Z, the set of integers (where Z had a second line through it). Read up on this for common set notations: https://www.mathsisfun.com/sets/symbols.html (toward the bottom).  We also briefly delved into proving that the square root of 2 is irrational via the method of contradiction: we assumed that it was rational, so by definition it could be written as a fraction a/b with a,b integers with no common factors besides 1, and then showed that if this assumption were correct, we could deduce that a and b were in fact not in lowest terms, so it followed our initial assumption (the rationality of square root 2) was incorrect. Don't worry if this is hard to grasp: it was meant as a preview for how proving mathematical statements works, which I certainly am not expecting any of you to do. Think of it as supplemental material. Here is a good link to the proof in case you are super interested: https://www.math.utah.edu/~pa/math/q1.html We went on to show some interesting properties of multiples. For example, if a and b are multiples of c, then so are (a+b) and (a-b). We also showed why the sum of digits trick for testing divisibility by 9 works (can be found in the notes). We went on to define the prime and composite numbers: a natural number p is prime if and only if p > 1 and p has no (integer) factors besides 1 and p. A natural number c > 1 is composite if it is not prime, that is, it can be written as ab for two integers a,b where 1 < a,b < c. (1 is neither prime nor composite.) We also showed that any natural number n > 1 can be decomposed into its prime factorization, where all its factors are prime numbers. According to the Fundamental Theorem of Arithmetic, every positive integer can be decomposed into a unique prime factorization. Further, we showed another very nice mathematical proof for the claim that there are infinitely many prime numbers. This was also done by the method of contradiction, where we began by supposing that were were only k primes for some finite number k, then showed that we could construct another prime number such that there were actually k+1 primes, hence our initial assumption (the finitude of primes) was again false. This proof is due to Euclid (c. 300 BC). Here is a good link to check out: http://www.math.utah.edu/~pa/math/q2.html Lastly, I mentioned the Goldbach Conjecture, which states that every even number greater than 2 can be expressed as the sum of two prime numbers (in class I neglected the "even" part of the statement). This is a "conjecture" because nobody has yet to prove the claim, even though a counterexample has not been found. In case you want to learn more about it, here's an informative Wikipedia link: https://en.wikipedia.org/wiki/Goldbach%27s_conjecture
Student Difficulties:	Saw improvement in the homeworks, with more attention paid to detail--good! Continue to do so, and please, no answers without work! Also remember to try everything, especially if you don't know how to do it: that's how you really improve your problem solving skills.
Notes	Please hand-in your homework solutions, written in a consistent notebook, in the next class.