1. Chapter 5: Number Theory

2. Divisibility
A number a is divisible by a number b if there exists a k such that a = bk. If b divides a, we write b | a.
b is called a factor or divisor of a
a is called a multiple of b

3. Primes and Composites
A number greater than 1 that has only 1 and itself as factors is prime
Any number greater than 1 that is not prime is composite
So a prime has exactly 2 factors, a composite has more than 2

4. Fundamental Theorem of Arithmetic
Every composite number can be expressed in one and only one way as a product of primes

5. Search for New Primes Most recent prime was discovered in 2013
Mersenne primes are of the form 2n -1
Great Internet Mersenne Prime Search (GIMPS)

6. Cardinality of Primes
Euclid’s proof from 300 B.C.: suppose there is a finite list of primes and get a contradiction

7. Greatest Common Factor
The GCF (greatest common factor) of a group of natural numbers is the largest natural number that is a factor of all the numbers
Find the prime factorizations of each and take the intersection of the prime factors
Two numbers are relatively prime if their GCF is 1

8. Least Common Multiple
The LCM (least common multiple) of a group of natural numbers is the smallest number that is a multiple of all the numbers in the group.
Find the prime factorizations of each number and take the union of the factors
If two numbers are relatively prime, then their LCM is just the product of the two numbers