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Chapter 5

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

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Chapter 5

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  1. Chapter 5 Number Theory

  2. 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.

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

  4. 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.

  5. 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.

  6. Divisibility Tests Theorem: Let a, m, n and k be whole numbers where a. b. c.

  7. 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

  8. 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

  9. 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).

  10. 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.

  11. 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.

  12. Some More Theorems Theorem: If a and b are whole numbers, with , then Theorem: If a and b are whole numbers, with and then

  13. 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).

  14. 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.

  15. 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.

  16. Two More Theorems Theorem: Let a and b be any two whole numbers. Then Theorem: There is an infinite number of primes.

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