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Chapter 5. Number Theory. 5.1 Primes, Composites and Tests for Divisibility.

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chapter 5

Chapter 5

Number Theory

5 1 primes composites and tests for divisibility
5.1 Primes, Composites and Tests for Divisibility

Definition: A counting number with exactly two different factors is called a prime number, or simply, a prime. A counting number with more than two factors is called a composite number, or simply, a composite.

fundamental theorem of arithmetic
Fundamental Theorem of Arithmetic

Each composite number can be expressed as the product of primes in exactly one way.

divides
Divides

Let a and b be any whole numbers with a not equal to 0. We say that a divides b, and write if and only if there is a whole number x such that

The symbol means that a does not divide b.

equivalent statements
Equivalent Statements

The following statements are equivalent.

  • a divides b.
  • a is a divisor of b.
  • a is a factor of b.
  • b is a multiple of a.
  • b is divisible by a.
divisibility tests
Divisibility Tests

Theorem: Let a, m, n and k be whole numbers where

a.

b.

c.

two more theorems
Two More Theorems

Theorem: A number is divisible by the product ab of two nonzero whole numbers a and b if it is divisible by both a and b and a and b have only the number 1 as a common factor.

Theorem: To test for prime factors of a number n, one need only search for prime factors p of n where

5 2 counting factors gcf and lcm
5.2 Counting Factors, GCF and LCM

Theorem: Suppose that a counting number n is expressed as a product of distinct primes with their respective exponents.

Then the number of factors of n is the product

greatest common factor
Greatest Common Factor

Definition: The greatest common factor (GCF) of two or more nonzero whole numbers is the largest whole number that is a factor of both (all) the numbers. The GCF of a and b is written GCF(a,b).

methods of finding gcf
Methods of Finding GCF

Set Intersection Method: Find the gcf(24, 36).

Step 1: List all factors of 24 and of 36.

Step 2: Find the intersection.

Step 3: Pick out the largest common factor.

methods of finding gcf1
Methods of Finding GCF

Prime Factorization MethodFind the gcf(24, 36).

Step 1: Write 24 and 36 as products of primes.

Step 2: Write each factor that appears in both factorizations.

Step 3: Use the smallest exponent that appears in either factorization.

some more theorems
Some More Theorems

Theorem: If a and b are whole numbers, with , then

Theorem: If a and b are whole numbers, with and

then

least common multiple
Least Common Multiple

Definition: The least common multiple (LCM) of two or more nonzero whole numbers is the smallest nonzero whole number that is a multiple of each (all) the numbers. The LCM of a and b is written LCM(a,b).

slide14

Methods of Finding LCM

Set Intersection Method: Find the lcm(24, 36).

Step 1: List a few multiples of 24 and of 36.

Step 2: Find the intersection.

Step 3: Pick out the smallest common multiple.

slide15

Methods of Finding LCM

Prime Factorization MethodFind the lcm(24, 36).

Step 1: Write 24 and 36 as products of primes.

Step 2: Write each factor that appears in either factorization.

Step 3: Use the largest exponent that appears in either factorization.

two more theorems1
Two More Theorems

Theorem: Let a and b be any two whole numbers. Then

Theorem: There is an infinite number of primes.