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Chapter 4 Number Theory

Activity: Rectangle Dimensions. 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. A natural number that has exactly two distinct factors is called a prime number. A natural number that has more than two distinct factors is called a composite number.Note: The number 1 has only one distinct factor, so it is neither prime nor composite..

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Chapter 4 Number Theory

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    1. Chapter 4 Number Theory Section 4.2 Prime and Composite Numbers

    2. Activity: Rectangle Dimensions 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.

    3. Comparing Prime and Composite Numbers A natural number that has exactly two distinct factors is called a prime number. A natural number that has more than two distinct factors is called a composite number. Note: The number 1 has only one distinct factor, so it is neither prime nor composite.

    4. Activity: Identifying Prime Numbers Sieve of Erastosthenes: used to identify prime numbers Using the list of numbers from 1-100, cross out all multiples of 2 (except 2 itself). Do the same for multiples of 3, 5, and 7. What do you discover about the numbers that remain?

    5. Using the Calculator to Determine if Numbers are Prime To determine if larger numbers are prime, take the square root of the number, then check to see if those prime numbers less than or equal to the square root are factors. (Its only necessary to check the prime numbers as factors because all other whole numbers are multiples of primes.) Example: Determine if 367 is prime or composite.

    6. Fundamental Theorem of Arithmetic Sometimes referred to as the Unique Factorization Theorem. The Fundamental Theorem of Arithmetic states that each composite number can be expressed as the product of prime numbers in exactly one way, disregarding the order of the factors.

    7. Examples 1.) Find the prime factorization of 84 using: a.) factor tree b.) stacked division 2) Find the prime factorization of 150.

    8. Greatest Common Factor The greatest common factor (GCF) of two natural numbers is the greatest natural number that is a factor of both numbers.

    9. 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. 3.) Determine the GCF of 60 and 140 using the Euclidean algorithm.

    10. Least Common Multiple The least common multiple (LCM) of two natural numbers is the smallest natural number that is a multiple of both the natural numbers.

    11. Examples 1.) 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.

    12. Important Reminders When using prime factorization to find the GCF and LCM: The GCF is found by using the minimum exponent for each shared prime power. The LCM is found by using the maximum exponent for each prime power.

    13. Twin Primes and Relative Primes Twin primes: any two consecutive primes that differ by 2 Examples: 3, 5 5, 7 11, 13 Two numbers a and b are relatively prime iff GCF (a,b) = 1.

    14. The GCF-LCM Product Theorem The product of the GCF and the LCM of two numbers is the product of the two numbers. Example: Find the GCF and LCM of 12 and 16. Compare the product of the GCF and LCM to the product of the numbers.

    15. Common Multiples and Common Factors Suppose S = NA where N is a whole number. We say that S is a multiple of A (and a multiple of N) and that N and A are factors of S. How can we recognize that we are concerned with multiples? In terms of the meaning of multiplication/division, S is the total of N groups of size A. So if we want to find total 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 total that can be formed into a certain number of groups. Developing Conceptual Understanding for Teaching Elementary Mathematics, University of Alabama

    16. Common Multiples and Common Factors How can we recognize that we are concerned with factors? In terms of the meaning of multiplication/division, if we are starting with a total (S) and looking for the size of a group (A) or the number of groups (N), we are looking for factors.

    17. 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. Then solve the problem, describing your reasoning.

    18. 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?

    19. Activity: Problem 1 Solution We are going to buy several boxes of pencils and several boxes of erasers. Hence, we are looking at groups of size 12 and groups of size 18. We want the total number of pencils and the total number of erasers to match up. The total number in combining groups of a certain size implies multiplication. Therefore we are looking for multiples of 12 and multiples of 18. We can list multiples of each until one multiple of 12 gives the same total as another multiple of 18.

    20. Activity: Problem 1 Solution

    21. Activity: Problem 1 Solution We see that if we buy 3 boxes of pencils and 2 boxes of erasers, we will have a total of 36 pencils and 36 erasers; so, we can make complete sets. If we buy 6 boxes of pencils and 4 boxes of erasers, we will have a total of 72 pencils and erasers so we can make complete sets. (We could of course buy more boxes if we need more sets.) The smallest number of total pencils (erasers) which we need in order to make complete sets is 36. This is the LEAST COMMON MULTIPLE of 12 and 18.

    22. Activity: Common Multiples and Common Factors 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, 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?

    23. Activity: Problem 2 Solution Patty wants to divide up her total collection of each type of candy into groups of the same size. We want to know the number of groups (and the size of the group). We are thus finding common factors.? We can proceed by finding all the factors of each number and looking for common factors. Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24 Factors of 16: 1, 2, 4, 8, 16 Developing Conceptual Understanding for Teaching Elementary Mathematics, University of Alabama

    24. Activity: Problem 2 Solution Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24 Factors of 16: 1, 2, 4, 8, 16 We see that 2, 4, and 8 are common factors. For the factor 2: Patty can divide each type of candy into packets with 2 pieces in each packet. She will then have 12 packets of gum drops and 8 packets of bubble gum. For the factor 4: Patty can divide each type of candy into packets with 4 pieces in each packet. She will then have 6 packets of gum drops and 4 packets of bubble gum. For the factor 8: Patty can divide each type of candy into packets with 8 pieces in each packet. She will then have 3 packets of gum drops and 2 packets of bubble gum. Here 8 is the GREATEST COMMON FACTOR of 24 and 16, i.e. it is the largest number that is a factor of each of the numbers. Developing Conceptual Understanding for Teaching Elementary Mathematics, University of Alabama

    25. 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

    26. Activity: Problem 3 Solution Even though the wording uses the word divide, what we are looking for is a total number of pieces. We want that number of pieces to be a multiple of 8 (since we will be dividing it into 8 groups) and be a multiple of 6 (since we will be dividing it into 6 groups). Hence, we are looking for a common multiple of 8 and 6. If we want each piece to be large, well want the total number of pieces to be small. Hence, we are looking for the least common multiple. 6: 6, 12, 18, 24 8: 8, 16, 24 Our least common multiple is 24. We will divide the candy bar into 24 pieces. (When it is shared by 8 children, each child will get 3 pieces; when it is shared by 6 children, each child will get 4 pieces.)

    27. Recap LEAST COMMON MULTIPLE of numbers: The smallest number that is a multiple of each of the given numbers. GREATEST COMMON FACTOR of numbers: The largest number that is a factor of each of the given numbers.

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