Chapter 4 Number Theory Activity: Rectangle Dimensions y Section 4.2, page 221 Prime and Composite Numbers NUMBER DIMENSIONS OF RECTANGLES FACTORS PRIME or COMPOSITE? Using the Dimensions of Rectangles Chart, use Snap-Cubes to create as many rectangles as possible using the number of cubes listed in the left-hand column. Comparing Prime and Composite Numbers 12 A natural number that has exactly two distinct factors is called a _________ number. y A natural number that has more than two distinct factors is called a _________ number. y Note: The number 1 has only one distinct factor, so it is _________ prime nor composite. 9 y 10 2 5 7 Activity: Identifying Prime Numbers Sieve of Erastosthenes: used to identify prime numbers y Using the list of numbers from 1100, 00, cross c oss out all a multiples u t p es of o 2 (except 2 itself). y Do the same for multiples of 3, 5, and 7. y What do you discover about the numbers that remain? y SIEVE of E R A S T O S T H E N E S 1 Using the Calculator to Determine if Numbers are Prime y To determine if larger numbers are prime, take the ________ _______ of the number, then check to see if those _________ numbers less than or equal to the square root are factors. (It’s only necessary to check the prime numbers as factors because all other whole numbers are multiples of primes.) Fundamental Theorem of Arithmetic Sometimes referred to as the Unique Factorization Theorem. y The Fundamental Theorem of Arithmetic t et c states tthat at each eac composite number can be expressed as the _________ of prime numbers in exactly one way, disregarding the order of the factors. y Examples 1) Find the prime factorization of 84 using: b) stacked division Using the Calculator to Determine if Numbers are Prime y Example: Determine if 367 is prime or composite. Examples 1) Find the prime factorization of 84 using: a) factor tree Examples 2) Find the prime factorization of 150. 2 Greatest Common Factor, page 229 Example of Euclidean Algorithm Find the GCF of 253 & 322. The greatest common factor (GCF) of two natural numbers is the greatest natural number that is a factor of both numbers. numbers y Three Methods for Finding GCF 1. Prime Factorization 2. _________ Diagram 3. Euclidean _________ (pg. 232) y Examples Examples 1) Determine the GCF of 84 and 150 by using the prime factorization of each. 2) Determine the GCF of 24 and 32 by listing the factors and using set notation in conjunction with a Venn diagram. g Examples Least Common Multiple 3) Determine the GCF of 60 and 140 using the Euclidean algorithm. y y The least common multiple (LCM) of two natural numbers is the _________ natural number that is a multiple p of both the natural numbers. 3 Examples 1) y Examples Use the intersection of sets to find the LCM of 3 and 8. 2) Use prime factorization to find the LCM of 9 and 12. Important Reminders Twin Primes and Relative Primes When using prime factorization to find the GCF and LCM: ◦ The GCF is found by using the _________ exponent for each shared prime power. ◦ The LCM is found by using the _________ exponent for each prime power. y Twin primes: any two _________ primes that differ by 2 Examples: 3, 5 5, 7 11, 13 y Two numbers a and b are _________ prime iff GCF (a,b) = 1. The GCF GCF--LCM Product Theorem The product of the GCF and the LCM of two numbers is the product of the two numbers. y Example: a p e: Find d the t e GCF GC and a d LCM C of o 12 and 16. Compare the product of the GCF and LCM to the product of the numbers. Common Multiples and Common Factors y y y Suppose S = NA where N is a whole number. We say that S is a _________ of A (and a multiple of N) and that N and A are _________ of S. How can we recognize that we are concerned with multiples? In terms of the meaning g of multiplication/division, S is the total of N groups of size A. So if we want to find _______ of a number of groups of a certain size (repeating groups of the same size), we are looking for a multiple of that size. Also, you can recognize finding multiples if you are asked to find a ______ that can be formed into a certain number of groups. Developing Conceptual Understanding for Teaching Elementary Mathematics, University of Alabama 4 Common Multiples and Common Factors y How can we recognize that we are concerned with factors? In terms of the meaning of multiplication/division, if we are starting sta t g with w t a total tota (S) and a d looking oo g for the size of a _________ (A) or the number of _________(N), we are looking for factors. Developing Conceptual Understanding for Teaching Elementary Mathematics, University of Alabama Activity: Common Multiples and Common Factors For each of the following three problems, read carefully and first explain if you are looking for multiples or factors and how you know. y Then solve the problem, describing your reasoning. y Developing Conceptual Understanding for Teaching Elementary Mathematics, University of Alabama Activity: Common Multiples and Common Factors Activity: Common Multiples and Common Factors 1) Pencils are packaged 12 pencils to a box. Erasers are packaged 18 to a box.You wish to buy just enough packages of each so that you can form complete sets of one pencil and one eraser (with nothing left over). How many packages of each will you buy? How many complete sets of erasers and pencils will you have? 2) Patty is making up candy packets. She has a box containing 24 gum drops and a bag containing 16 pieces of bubble gum. She want to divide up the candy into packets so that there is only one kind of candy in a packet, packet each packet contains the same number of pieces, and all the candy is used up. Suppose further that she wants as many pieces in each packet as possible. How many pieces of candy are in each packet? How many packets can she make up? Developing Conceptual Understanding for Teaching Elementary Mathematics, University of Alabama Activity: Common Multiples and Common Factors 3) You have a large candy bar. You want to divide it into equal size pieces so that if 8 children wanted to share the candy bar equally, each child would get a whole number of pieces and if 6 children wanted to share the candy bar equally, each child would get a whole number of pieces. How many pieces would you need to divide the candy bar into, assuming you wanted the pieces to be as large as possible? Developing Conceptual Understanding for Teaching Elementary Mathematics, University of Alabama Recap y LEAST COMMON MULTIPLE of numbers: The smallest number that is a multiple of each of the given numbers. y GREATEST COMMON FACTOR of numbers: The largest number that is a factor of each of the given numbers. Developing Conceptual Understanding for Teaching Elementary Mathematics, University of Alabama 5