Thinking Mathematically

Slides:



Advertisements
Similar presentations
Preview Warm Up California Standards Lesson Presentation.
Advertisements

Prime Factorization Notes 9/25/09
Chapter 5 Number Theory © 2008 Pearson Addison-Wesley. All rights reserved.
Do Now 11/30/09 Copy HW in your planner. Copy HW in your planner. –Text page 175, #16-44 multiples of 4; #48-54 evens Be ready to copy POTW #3 Be ready.
Prime and Composite Numbers
Thinking Mathematically
Preparation for NS2.4 Determine the least common multiple and the greatest common divisor of whole numbers; use them to solve problems.
PRIME FACTORIZATION USING FACTOR TREES!.
5.1 Number Theory. The study of numbers and their properties. The numbers we use to count are called the Natural Numbers or Counting Numbers.
Number Theory and the Real Number System
1 Section 2.4 The Integers and Division. 2 Number Theory Branch of mathematics that includes (among other things): –divisibility –greatest common divisor.
5 Minute Check Complete in your notebook.
Prime Factorization: Writing a composite number as a product (multiplication problem) of prime numbers 1.) Use a factor trees to the find the prime factorization.
The Fundamental Theorem of Arithmetic (2/12) Definition (which we all already know). A number greater than 1 is called prime if its only divisors are 1.
Fractions and Rational Expressions
Preparation for NS2.4 Determine the least common multiple and the greatest common divisor of whole numbers; use them to solve problems with fractions (e.g.
Chapter Number Theory 4 4 Copyright © 2013, 2010, and 2007, Pearson Education, Inc.
BY MISS FARAH ADIBAH ADNAN IMK
5 Minute Check Complete in your notebook.
A number that divides evenly into a larger number without a remainder Factor- Lesson 7 - Attachment A.
Section 5.1 Number Theory.
1 Properties of Integers Objectives At the end of this unit, students should be able to: State the division algorithm Apply the division algorithm Find.
Holt CA Course Prime Factorization Preparation for NS2.4 Determine the least common multiple and the greatest common divisor of whole numbers; use.
Prime Numbers and Prime Factorization. Factors Factors are the numbers you multiply together to get a product. For example, the product 24 has several.
Numbers MST101. Number Types 1.Counting Numbers (natural numbers) 2.Whole Numbers 3.Fractions – represented by a pair of whole numbers a/b where b ≠ 0.
SECTION 5-1 Prime and Composite Numbers Slide
Factors
5.1 Divisibility. Natural Numbers The set of natural numbers or counting numbers is {1,2,3,4,5,6,…}
Factors are numbers you can multiply together to get another number. Multiples are numbers that can be divided by another number without a remainder. Its.
Factors, Primes & Composite Numbers 6 th grade math.
Copyright © 2009 Pearson Education, Inc. Chapter 5 Section 1 - Slide 1 Chapter 1 Number Theory and the Real Number System.
Prime Factorization Notes 9/8/10 Composite Number- a whole number greater than 1 with more than two factors. Prime Number – A whole number with exactly.
Copyright © Cengage Learning. All rights reserved. 1 Whole Numbers.
5 Minute Check Complete on the back of your homework. Tell whether each number is divisible by 2,3,4,5,6,9, ,681.
Holt CA Course Prime Factorization Warm Up Warm Up California Standards California Standards Lesson Presentation Lesson PresentationPreview.
Slide Copyright © 2009 Pearson Education, Inc. Unit 1 Number Theory MM-150 SURVEY OF MATHEMATICS – Jody Harris.
Factor A factor of an integer is any integer that divides the given integer with no remainder.
Factors and Prime Factorization
Factors & Number Theory
Slide Copyright © 2009 Pearson Education, Inc. Slide Copyright © 2009 Pearson Education, Inc. Chapter 1 Number Theory and the Real Number System.
 2012 Pearson Education, Inc. Slide Chapter 5 Number Theory.
© 2010 Pearson Prentice Hall. All rights reserved. CHAPTER 5 Number Theory and the Real Number System.
Slide Copyright © 2009 Pearson Education, Inc. 5.1 Number Theory.
1 Discrete Structures – CNS2300 Text Discrete Mathematics and Its Applications Kenneth H. Rosen (5 th Edition) Chapter 2 The Fundamentals: Algorithms,
Do Now Write as an exponent 3 x 3 x 3 x 3 4 x 4 x 4 x 4 x 4 5 x 5 x 5 What is a factor? Define in your own words.
Number Theory: Prime and Composite Numbers
One is Special! Click to Play Primes, Factors, & Multiples Factor a whole number that divides into another whole number without remainder Product the.
Slide Copyright © 2009 Pearson Education, Inc. Slide Copyright © 2009 Pearson Education, Inc. Chapter 1 Number Theory and the Real Number System.
FACTOR Definition: A number that divides evenly into another number Clue: Factors of 6 are 1, 2, 3, 6.
Factors, Primes & Composite Numbers Chapter 4.1. Definition  Product – An answer to a multiplication problem. 7 x 8 = 56 Product.
Section 5.1 Number Theory.
Factors
Number Theory.
Number Theory and the Real Number System
Chapter 5: Number Theory
Copyright © 2016, 2013, and 2010, Pearson Education, Inc.
Thinking Critically 4.1 Divisibility Of Natural Numbers
Exercise 24 ÷ 2 12.
Prime Factorization Prime factorization is the long string of factors that is made up of all prime numbers.
Section 5.1 Number Theory.
Number Theory and the Real Number System
A number that divides evenly into a larger number without a remainder
Chapter 5 Number Theory 2012 Pearson Education, Inc.
Prime Factorization Course
Section 5.1 Number Theory.
Prime Numbers and Prime Factorization
Chapter 5 Number Theory 2012 Pearson Education, Inc.
Copyright © 2013, 2010, and 2007, Pearson Education, Inc.
Prime Factorization FACTOR TREE.
Number Theory: Prime & Composite Numbers
Presentation transcript:

Thinking Mathematically Number Theory: Prime and Composite Numbers

The Set of Natural Numbers

Divisibility If a and b are natural numbers, a is divisible by b if the operation of dividing a by b leaves a remainder of 0. This is the same as saying that b is a divisor of a, or b divides a. All three statements are symbolized by writing b|a.

Discuss the Rules of Divisibility [p. 194] Discuss using a calculator to check divisibility

Prime Numbers A prime number is a natural number greater than 1 that has only itself and 1 as factors.

Composite Numbers A composite number is a natural number greater than 1 that is divisible by a number other than itself and 1.

The Fundamental Theorem of Arithmetic Every composite number can be expressed as a product of prime numbers in one and only one way (if the order of the factors is disregarded). For example 700 can be written in the following way. 700 = 2 x 2 x 5 x 5 x 7 The prime factors of 700 are 2, 5, and 7.

“Factor Trees” The prime factors of a natural number can be found by constructing a “factor tree.” Write the given number as a product and continue to factor each composite number until only prime numbers remain. The “prime factorization” of 40 is determined by the prime numbers at the bottom of each branch of the tree. 40 x 8 5 4 2 x 40 = 2 x 2 x 2 x 5 = 23 x 5 2 x The prime factors of 40 are 2 and 5.

Finding the Greatest Common Divisor of Two or More Numbers Using Prime Factorization To find the greatest common divisor of two or more numbers: Write the prime factorization of each number. Select each prime factor with the smallest exponent that is common to each of the prime factorizations. Form the product of the numbers from step 2. The greatest common divisor is the product of these factors.

Example of Finding the Greatest Common Divisor 40 x 8 5 4 2 24 x 6 4 3 2 x 2 x 40 = 2 x 2 x 2 x 5 24 = 2 x 2 x 2 x 3 The greatest common divisor of 24 and 40 is 2 x 2 x 2 = 8.

Finding the Least Common Multiple Using Prime Factorization To find the least common multiple of two or more numbers: Write the prime factorization of each number. Select every prime factor that occurs, raised to the greatest power to which it occurs, in these factorizations. Form the product of the numbers from step 2. The least common multiple is the product of these factors.

Example of the Least Common Multiple 40 x 8 5 4 2 12 x 3 4 2 40 = 2 x 2 x 2 x 5 12 = 2 x 2 x 3 The least common multiple of 12 and 40 is 2 x 2 x 2 x 3 x 5 = 120.

Thinking Mathematically Number Theory: Prime and Composite Numbers