Prime Numbers A prime number is a whole number, greater than 1, that has only 1 an itself as factors. Composite Numbers A composite number is a whole number, greater than 1, that are not prime. Prime Factorization To find the prime factorization of a whole number means to write it as the product of only prime numbers. This is can be useful when finding things like the Greatest Common Factor or Least Common Multiple between two numbers (we’ll get into that later). Example: Factor 90 into its prime factors. Choose any two factors of 90 (besides 1 and 90) Then do the same with each of those factors. Keep going until you have only prime factors as the bottom “roots” of the “factor tree.” 90 9 3 10 3 2 5 90 = 3 3 2 5 ● ● ● Putting these factors in numerical order and then combing like terms into exponents gives: 90 = 2●32●5 Theorem: Any composite number has exactly one set of prime factors. Example 5 Find the prime factorization of 210 First, pick any two factors of 210. For instance 21 and 10. We could have also picked 7 and 30 as the factors. 210 21 3 210 10 7 2 7 30 5 Notice that either method gives us 210 = 2●3●5●7 6 3 5 2 Alternate Factoring Method – I call it the “Pyramid Division Method” For more info: http://www.purplemath.com/modules/factnumb.htm With this method, you start and the bottom of the pyramid and move up, so you have to leave lots of room at the top of your problem. Example: Find the prime factorization of 90. Start with the first prime factor you can think of that goes into 90. We can’t choose 9 or 10 because they aren’t prime. Since 90 is even, start with the factor 2. 3 39 5 45 is 3 prime? Yes! You are done. is 9 prime? No. Keep dividing. is 45 prime? No. Keep dividing. 2 90 Prime Factorization of 90 is 2x5x3x3 Write the factors in numerical order and use exponents where factors are repeated = 2 32 5 Upside-Down Division Method Example: Find the prime factorization of 60 2 60 3 30 Is 30 prime? No. 2 10 Is 10 prime? No. 5 Prime Factorization of 60 is 22 3 5 Is 5 prime? Yes. Stop Greatest Common Factor The Greatest Common Factor is the biggest number that can be divided into two ore more numbers. It is also called the greatest common divisor. It is called greatest common factor because factor of a number is a factor is a number that another number is divisible by. Even though is called the “Greatest”, it cannot be bigger than the smaller of the two numbers, since larger numbers could not be divided into it. 4 is the Greatest Common Factor of 12 and 16 because it is the biggest number that 12 is divisibly by and 16 is divisible by. 12 is the Greatest Common Factor of 12 and 48 because 12 is the biggest number that 12 is divisible by and that 48 is divisible by. Another way to do this is the Birthday Cake Method. Multiply the numbers on the left side of the birthday cake. In the example, multiply 2 x 2 x 3. This will give you a GCF o LEAST COMMMON MULTIPLE Here’s a funny video clip: http://mathdogmedia.blogspot.com/2013/01/lcm-father-of-bride.html The least common multiple is the smallest multiple that is in common between 2 or more numbers. What are Multiples? Multiples are products of a number and another number. The multiples of 2 are: 2, 4, 6, 8, 10, 12, 14, 16, …. it goes on forever! The multiples of 3 are : 3, 6, 9, 12, 15, 18, 21, 24, …. The multiples of 4 are: 4, 8, 12, 16 20, 24, 28, ….. And so on… What is the LEAST COMMON MULTIPLE of 12 and 48? Multiples of 12: 12, 24, 36, 48, … Multiples of 48: 48, 96,… The first multiple in common is 48, so the LCM is 48. Birthday Cake Method: Multiply all of the numbers on the outside of the birthday cake, going do wn the left side and across the bottom. Follow the “L” for LCM. In the example, multiply 2 x 2 x 3 x 1 x 4. So, the LCM is 48. Finding the LCM by Prime Factorization Step 1: Find the prime factorization of each number (use a factor tree if needed) Step 2: Write down each of the prime factors without their exponents. Step 3: Choose the highest exponent for each prime factor. Step 4: Multiply the prime factors in the line LCM = ______________ = ____ And writeh that product after the second equal sign. Example: Find the LCM of 8 and 12 Step 1: Prime Factorization of 8: 2x2x2 = 23 Prime Factorization of 12: 2x2x3 = 22x3 Step 2: Write down the prime factors. The only ones are 2 and 3. 2 x 3 Step 3: Choose the highest exponent of each prime factor. The highest exponent of the 2’s is 3 and the highest exponent of the 3’s is 1 (3 = 31) Step 4: LCM = 23 x 31 = 24 Method 3: Find the LCM using the Division Method. Recall the division method for finding the GCF. Find the GCF of 12 and 30: Divide both numbers by the first prime factor you can think of that goes into both 12 and 30. 2 12 3 30 6 15 2 5 We stop here because 2 and 5 are relatively prime. For the GCF only circle the factors “outside” the boxes. GCF of 12 and 30 = 2x3 = 6 For the LCM we multiply the GCF by the relatively prime factors in the last box. LCM = 6 x 2 x 5 = 60 Greatest Common Factor and Least Common Multiple Word Problems 1. A math teacher and a science teacher combine their first period classes for a group activity. The math class has 24 students and the science class has 16 students. The students need to divide themselves into groups of the same size. Each group must have the same number of math students. Find the greatest number of groups possible. 2. A photography club is practicing developing techniques. One set of negatives contains 32 negatives and another contains 28 negatives. Each set can be divided equally among the members present. List all the possible numbers of members present. What is the greatest possible number? 3. Organizers for a middle school culmination have set up chairs in two sections. They put 126 chairs for graduates in the front section and 588 chairs for guests in the back section. If all rows have the same number of chairs, what is the greatest number of chairs possible? 4. A quality control inspector in an egg factory checks every 36th egg for cracks and every 42nd egg for weight. What is the number of the first egg each day that the inspector checks for both qualities? (Hint: think this will be the least number....) 5. The manager at Frank's Snack Shack buys hot dogs in packages of 30. He buys hot dog buns in packages of 24. Unfortunately, he cannot buy part of a package. What is the least number of packages of each product he can buy to have an equal number of hot dogs and hot dog buns? (Careful: What is the question asking for??) 6.Pepe enjoys bird-watching and observed two types of birds traveling this season: ducks and seagulls. While the ducks traveled in flocks of 15, the seagulls traveled in flocks of 12. If Pepe observed the same total number of ducks and seagulls, what is the smallest number of ducks that he could have observed?
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