Class Summary - 10/16/16

Instructors:	Mr. Rameel Rizvi
Attendance:	Aiden, Omar, Areej, AbdulRafie
Homework:	Read: Chapter 3.8 (this is a review of all main concepts from the chapter--do this slowly and thoroughly); Problems to solve:  3.47, 3.55, 3.56, 3.60, 3.67, 3.71, 3.78, 3.79, 3.82, 3.85, 3.87  Note: start these early--they are not trivial
Challenge Problem:	Find the number of five-digit positive integers, n, that satisfy the following conditions: (a) the number n is divisible by 5; (b) the first and last digits of n are equal; and       (c) the sum of the digits of n is divisible by 5. (Source: AIME I 2013)
Class Activity:	Finished Chapter 3.
Concepts:	Defined divisibility and showed that a divides b (written a|b) iff b is a multiple of a (that is, we can write b = k*a for some integer k). "Iff" is shorthand for "if and only if," which means that either clause being true implies that the other is true. For example, A iff B when, if A is true then B is true and when B is true then A is true. Thus, in this case, if we ever know that m|n, then we also know that n is a multiple of m, and if we ever know that n is a multiple of m, then we know that m|n. Covered the concepts of least common multiple (LCM) and greatest common divisor (GCD), and showed that they are directly related to the prime factorizations of the numbers in question. To find the LCM of a set of numbers, we take all prime factorizations of the numbers in the set and construct the smallest number which is divisible by each such factorization; analogously, for the GCD of a set of numbers, we simply take all prime factorizations of the numbers and "intersect" them such that we obtain the largest number common to all the prime factorizations.
Student Difficulties:	Concepts of number theory, and definitions in general, seem to be hard to reason with. Again, this is to be expected given the stark contrast between the course material and what is traditionally covered in schools. It is important that we continue emphasizing the use of claims we have already proven to generate new ones, and this is done purely through insight. Always think, and if you ever get stuck: keep thinking.
Notes	Please hand-in your homework solutions, written in a consistent notebook, in the next class.