Chapter 5 Number Theory Divisibility A number a

Chapter 5: Number Theory

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

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

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

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

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

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

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