5.1

1. 5.1 Number Theory

2. Number Theory • The study of numbers and their properties. • The numbers we use to count are called natural numbers, N, or counting numbers.

3. 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. 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. 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. Rules of Divisibility Divisible by Test Example 2 The number is even. 846 3 The sum of the digits of the number is divisible by 3. 846 since 8 + 4 + 6 = 18 4 The number formed by the last two digits of the number is divisible by 4. 844 since 44  4 5 The number ends in 0 or 5. 285

7. Create a list from 1 – 100

8. 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. 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. Example of branching method 3190 319 10 11 29 2 5 Therefore, the prime factorization of 3190 = 2 • 5 • 11 • 29.

11. Division Method 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.

12. 3 663 13 221 17 Example of division method • 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

13. Example 1: p. 218# 37

14. 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. 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. Example (GCF) • Find the GCF of 63 and 105. 63 = 32• 7 105 = 3 • 5 • 7 • Smallest exponent of each factor: 3 and 7 • So, the GCF is 3 • 7 = 21.

17. Find the GCF between 36 and 54

18. 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. 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. Example (LCM) • Find the LCM of 63 and 105. 63 = 32 • 7 105 = 3 • 5 • 7 • Greatest exponent of each factor: 32, 5 and 7 • So, the LCM is 32 • 5 • 7 = 315.

21. Find the LCM between 36 and 54

22. Example of GCF and LCM • Find the GCF and LCM of 48 and 54. • Prime factorizations of each: 48 = 2 • 2 • 2 • 2 • 3 = 24• 3 54 = 2 • 3 • 3 • 3 = 2 • 33 GCF = 2 • 3 = 6 LCM = 24 • 33 = 432

23. Homework • P. 218# 15 – 54 (x3)