Chapter 5 The Real Number System Section 5-1 The Natural Numbers Objectives: Find the factors of a natural number Identify prime and composite numbers Find the prime factorization of a number Find the GCF of two or more numbers Find the least common multiple of two or more numbers

Prime and Composite Numbers The set of natural numbers consists of the numbers 1, 2, 3, 4, … Every natural number can be written as the product of two or more natural numbers. These numbers are called factors. A natural number a is divisible by a natural number b if dividing a by b results in a remainder of zero. A natural number is called prime if it has exactly two factors, 1 and itself. A natural number is called composite if it has three or more factors.

Example 1: Finding Factors Find all factors of 24. Find all factors of 50.

Example 2: Deciding if a Number is Prime Decide whether each number is prime or composite. (a)25 (b)17 (c)12 (d)31 (e)34 (f)29

Prime Factorization Every composite number can be written as the product of prime numbers, and there’s only one way to do so.

Example 3: Finding Prime Factorization Using the Tree Method Find the prime factorization of 100 using the tree method. Find the prime factorization of 360.

Greatest Common Factors The greatest common factor (GCF) of two or more numbers is the largest number that is a factor of all of the original numbers. How to Find the GCF of Two or More Numbers (1)Write the prime factorization of each number. (2)Make a list of each prime factor that appears in all prime factorizations. For prime factors with exponents, choose the smallest power that appears in each. (3)The GCF is the product of the numbers you listed in step 2.

Example 5: Find the GCF of Two Numbers Find the GCF of 72 and 180. Find the GCF of 54 and 144.

Example 6: Finding the GCF of Three Numbers Find the GCF of 40, 60, and 100. Find the GCF of 45, 75, and 150.

Example 7: Applying the GCF to Packaging Goods for Sale An enterprising student gets a great deal on slightly past-their-prime packets of instant coffee. He acquires 200 packets for decaf and 280 packets of regular coffee. The plan is to package them for resale at a college fair so that there’s only one type of coffee in each box, and every box has the same number of packets. How can he do this so that each box contains the largest number of packets? (Of course, he wants to package all of the coffee he has.)

Extra Example 7 The enterprising student later buys unsold Halloween candy for resale. If he has 360 fun- size bags of peanut M&Ms and 420 bags of plain M&Ms, how should he package them to follow the same guidelines as before?

Least Common Multiples The least common multiple (LCM) of two or more numbers is the smallest number that is a multiple of each. How to Find the LCM of Two or More Numbers (1)Write the prime factorization of each number. (2)Make a list of every prime factor that appears in any of the prime factorizations. For prime factors with exponents, choose the largest that appears in any factorization. (3)The LCM is the product of the numbers you listed in step 2.

Example 8: Finding the LCM of Three Numbers Find the LCM of 24, 30, and 42. Find the LCM of… (a)40, 50 (b)28, 35, 49 (c)16, 24, 32

Example 9: Applying the LCM to Grocery Shopping Have you ever noticed how many hot dogs come in packages of 10, but most hot dog buns come in packages of 8? What’s the smallest number of packages you can buy of each so that you end up with the same number of hot dogs and buns?

Extra Example 9 After getting her first job out of college, Colleen vows to buy herself a new item of clothing every 15 days, and a new pair of shoes every 18 days just because she can. How long with it be until she buys both items on the same day?

Homework P. 208 #7-23 EOO, EOO, 93, 96