1. Slide 5 - 1 Copyright © 2009 Pearson Education, Inc. 5.1 Number Theory

2. Slide 5 - 2 Copyright © 2009 Pearson Education, Inc. Number Theory The study of numbers and their properties. The numbers we use to count are called natural numbers, N, or counting numbers.

3. Slide 5 - 3 Copyright © 2009 Pearson Education, Inc. Factors The natural numbers that are multiplied together to equal another natural number are called factors of the product. Example: The factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24.

4. Slide 5 - 4 Copyright © 2009 Pearson Education, Inc. Divisors If a and b are natural numbers and the quotient of b divided by a has a remainder of 0, then we say that a is a divisor of b or a divides b.

5. Slide 5 - 5 Copyright © 2009 Pearson Education, Inc. Prime and Composite Numbers A prime number is a natural number greater than 1 that has exactly two factors (or divisors), itself and 1. A composite number is a natural number that is divisible by a number other than itself and 1. The number 1 is neither prime nor composite, it is called a unit.

6. Slide 5 - 6 Copyright © 2009 Pearson Education, Inc. Rules of Divisibility 285The number ends in 0 or 5. 5 844 since 44  4 The number formed by the last two digits of the number is divisible by 4. 4 846 since 8 + 4 + 6 = 18 The sum of the digits of the number is divisible by 3. 3 846The number is even.2 ExampleTestDivisible by

7. Slide 5 - 7 Copyright © 2009 Pearson Education, Inc. Create a list from 1 – 100 12345678910 11121314151617181920 21222324252627282930 31323334353637383940 41424344454647484950 51525354555657585960 61626364656667686970 71727374757677787980 81828384858687888990 919293949596979899100

8. Slide 5 - 8 Copyright © 2009 Pearson Education, Inc. The Fundamental Theorem of Arithmetic Every composite number can be expressed as a unique product of prime numbers. This unique product is referred to as the prime factorization of the number.

9. Slide 5 - 9 Copyright © 2009 Pearson Education, Inc. Finding Prime Factorizations Branching Method:  Select any two numbers whose product is the number to be factored.  If the factors are not prime numbers, continue factoring each number until all numbers are prime.

10. Slide 5 - 10 Copyright © 2009 Pearson Education, Inc. Example of branching method Therefore, the prime factorization of 3190 = 2 5 11 29. 3190 31910 11 292 5

11. Slide 5 - 11 Copyright © 2009 Pearson Education, Inc. 1. Divide the given number by the smallest prime number by which it is divisible. 2.Place the quotient under the given number. 3.Divide the quotient by the smallest prime number by which it is divisible and again record the quotient. 4.Repeat this process until the quotient is a prime number. Division Method

12. Slide 5 - 12 Copyright © 2009 Pearson Education, Inc. Write the prime factorization of 663. The final quotient 17, is a prime number, so we stop. The prime factorization of 663 is 3 13 17 Example of division method 13 3 17 221 663

13. Slide 5 - 13 Copyright © 2009 Pearson Education, Inc. Example 1: p. 218# 37

14. Slide 5 - 14 Copyright © 2009 Pearson Education, Inc. Greatest Common Factor The greatest common factor (GCF) of a set of natural numbers is the largest natural number that divides (without remainder) every number in that set.

15. Slide 5 - 15 Copyright © 2009 Pearson Education, Inc. Finding the GCF of Two or More Numbers Determine the prime factorization of each number. List each prime factor with smallest exponent that appears in each of the prime factorizations. Determine the product of the factors found in step 2.

16. Slide 5 - 16 Copyright © 2009 Pearson Education, Inc. Example (GCF) Find the GCF of 63 and 105. 63 = 3 2 7 105 = 3 5 7 Smallest exponent of each factor: 3 and 7 So, the GCF is 3 7 = 21.

17. Slide 5 - 17 Copyright © 2009 Pearson Education, Inc. Find the GCF between 36 and 54

18. Slide 5 - 18 Copyright © 2009 Pearson Education, Inc. Least Common Multiple The least common multiple (LCM) of a set of natural numbers is the smallest natural number that is divisible (without remainder) by each element of the set.

19. Slide 5 - 19 Copyright © 2009 Pearson Education, Inc. Finding the LCM of Two or More Numbers Determine the prime factorization of each number. List each prime factor with the greatest exponent that appears in any of the prime factorizations. Determine the product of the factors found in step 2.

20. Slide 5 - 20 Copyright © 2009 Pearson Education, Inc. Example (LCM) Find the LCM of 63 and 105. 63 = 3 2 7 105 = 3 5 7 Greatest exponent of each factor: 3 2, 5 and 7 So, the LCM is 3 2 5 7 = 315.

21. Slide 5 - 21 Copyright © 2009 Pearson Education, Inc. Find the LCM between 36 and 54

22. Slide 5 - 22 Copyright © 2009 Pearson Education, Inc. Example of GCF and LCM Find the GCF and LCM of 48 and 54. Prime factorizations of each: 48 = 2 2 2 2 3 = 2 4 3 54 = 2 3 3 3 = 2 3 3 GCF = 2 3 = 6 LCM = 2 4 3 3 = 432

23. Slide 5 - 23 Copyright © 2009 Pearson Education, Inc. Homework P. 218# 15 – 54 (x3)