§5.2 Primes and Composites

§5.2 Primes and Composites §5.2 Primes and Composites - Start

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§5.2 Primes and Composites - Description

2/23/17. Review!. How . do we decide when a number is prime or composite?. How do we use factor trees?. How do we find and use all primes less than 100?. Today We’ll Discuss. What are the divisibility tests . ID: 554823 Download Presentation

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§5.2 Primes and Composites




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Presentations text content in §5.2 Primes and Composites

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§5.2 Primes and Composites

2/23/17

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Review!How do we decide when a number is prime or composite?How do we use factor trees?How do we find and use all primes less than 100?

Today We’ll Discuss

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What are the divisibility tests for 2,3,4,5,6,8,9,10?What about 7?

Review

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Test #1: Take the digits of the number in reverse order, from right to left, multiplying them successively by the digits 1,3,2,6,4,5 repeating this sequence of multipliers as long as necessary. Add the products. If the sum is divisible by 7, then so is the whole number!

Divisibility Test for 7

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Ex: Is 1603 divisible by 7?3(1) + 0(3) + 6(2) + 1(6) = 217 | 21 so yes!

Divisibility Test for 7

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Test #2: Remove the last digit, double it, subtract it from the truncated original number and continue doing this until only one digit remains. If this is 0 or 7, then the original number is divisible by 7.

Divisibility Test for 7

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Ex: Is 1603 divisible by 7?160 – 2(3) = 15415 – 2(4) = 7Yes!

Divisibility Test for 7

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a.) 1538b.) 7861c.) 639d.) 749

Divisibility Test for 7

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Definition: A natural number is a prime number when it has exactly two distinct factors: 1 and itself. (1 is not prime!)Definition: A natural number is a composite number when it has more than two distinct factors.

Definitions of Prime & Composite

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Remember that the multiplication of any two factors can be represented in an area problem.Asking students to take a 1 x ___ area model and make a new one is a great way to show when a small number is prime or composite.

Using the Area Model

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Using the Area Model

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Example

1) Use the area model to show all the different ways to factor 20.

2) Use the area model to discuss why 23 is a prime number.

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Let’s take a look at the prime numbers less than 100. Sieve of Eratosthenes!

List of Primes

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Every natural number greater than 1 is either a prime number, or can be written as the product of prime factors in exactly one way (if order does not matter).Definition: The product above is called the prime factorization of a number.

The Fundamental Theorem of Arithmetic

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Factor trees are used to keep track of factors in the process of finding a prime factorization.By the FToA, all prime factorizations of a given number are the same, so you can start however you like!

Factor Trees

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Example

Write the prime factorization of each number.

a) 24

b) 126

c)

225

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Example

Write the prime factorization of each number.

a) 5929

b) 3500

c) 3773

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It is always helpful to divide out as large of factors as you go along. This helps to shorten the process.Don’t forget that if you are attempting to pull out all prime factors, certain factors may appear multiple times.

Factor Tree Tips

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If you are asked only if a number is prime or composite, you only have a bit of work to do….Theorem: To test for prime factors of a number n, you need only check for prime factors that are less than or equal to the square root of n.

Testing only Prime or Composite

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Example

Tell whether each number is prime or composite.

a) 901

b) 223

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Conjecture: Every even integer greater than 2 can be written as the sum of two primes.

Goldbach’s Conjecture

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Pages 176-179#4, 6, 11, 14, 18, 20, 23, 28, 38, 41

Homework #11 - §5.2


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