Least Common Multiples Looking “Forward” with the Values

What is our objective? Today we will work with least common multiples (LCM). We will find the LCM for a set of numbers. We will also use the LCM of a set of numbers to solve word problems. Please remember……a common multiple is needed when working with unlike fractions!

Vocabulary Common multiples: Least common multiple: Common multiples are multiples that are shared by a set of numbers. (notes) The smallest, shared multiple…..

Why do we use the LCM? We use the LCM to create common denominators. Common denominators are needed to add and subtract fractions. We also use common denominators to solve word problems involving different values that must be distributed evenly. These word problems often involve unlike fractions!

First things first….finding the LCM The first method is the one You have probably used before. Step 1: List multiples of each number. 12: 12, 24, 36, 48, 60, 72, 84… 36: 36, 72, Step 2: Identify the common multiples Step 3: Find the least multiple shared 36 is the LCM

First things first….finding the LCM Step 1: List multiples of each number. 5: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55 …. 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, Step 2: Identify the common multiples Step 3: Find the least multiple shared 20 is the LCM

In our last example, 20 was the LCM. When the numbers you are working with do NOT share any common factors, the product of the numbers is the LCM. Here are a few examples of this. You may want to put these in your notes. 3 and 5 share no factors: LCM = 15 7 and 5 share no factors: LCM = 35 8 and 9 share no factors: LCM = and 9 share no factors: LCM = 90

Good to Know…. If you are looking for the LCM, and one number is a factor of another, just use the large number. For example, if you are looking for the LCM of 4 and 16, since 4 is found “inside” 16, 16 is the LCM Since 12 is a factor of 24, just use the 24 and 18 !! Since 6 is found “inside” 12, just use the 12 and 30 30

The second method is helpful when the values you are working with share factors. This method uses the prime factorization method. Let’s look at 12 and 20. Factors are shared by these numbers. 12: ( 6 x 2) = 2 x 3 x 2 20: ( 4 x 5) = 2 x 2 x 5 We now have a complete list of all factors, but we will not duplicate the shared factors. 2 x 3 x 2 x 5 = 60 LCM Here’s another way!! x 3 x 5 = 60

Because the second method is new, let’s try another example. Find the LCM for 18, 24, and : ( 6 x 3) = 2 x 3 x 3 24: ( 6 x 4) = 2 x 3 2 x 2 Multiply the “value” of each swim lane. This keeps you from duplicating any shared values. 2 x 3 x 3 x 2 2 x 5 = 30: ( 6 x 5) = 2 x is the LCM x 3 x 5 x 6 = 360 5

Let’s try another example. Three methods will be shown. Find the LCM for 12, 20, and 8 8: ( 4 x 2) = 2 x 2 x 2 20: ( 5 x 4) = 2 x 2 5 8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, 104, 112, : 20, 40, 60, 80, 100, : 12,24,36,48,60,72,84,96,108,120 2 x 2 x 2 x 5 x 3 = : ( 4 x 3) = 2 x 2 3

x2 x 3 x 5 = Be careful with this method. If you do not start with the GCF, your multiple will NOT be the LCM……… AND if the “pink” numbers also share a factor, they you will get a number that is too big!! x 3 2 x 2 X

Partner Practice For each number set, find the LCM. Try using each method ( 3 x 2) 3 x 2 (3 x 7) 3 x 7 LCM 42 6, 12, 18, 24, 30, 36, 42 21, 42 ( 4 x 2) 2 x 2 x 2 (5 x 2) 2 x 5 3 X 2 x 7 2 x 2 x 2 x 5 8, 16, 24, 32, 40 10, 20, 30, 40 LCM 40 These numbers share NO factors: 7 x 8 = 56 LCM 56

How is this used????? Consumer Application: Group Discussion English muffins come in packs of 8, and eggs in cartons of 12. If there are 24 students, what is the least number of packs and cartons needed so that each student has a muffin sandwich with one egg and there are no muffins left over? There are 24 English muffins and 24 eggs. Think of muffins in groups of 8. Think of eggs in groups of 12. We are looking for the smallest multiple they both share. They have given us the LCM of 24!!! We are being asked, “What multiple of 8 is 24?” 8n = 24? So 3 packs of English muffins are needed. They have given us the LCM of 24!!! We are being asked,“What multiple of 12 is 24?” 12n = 24? So 2 cartons of eggs are needed.

Here is another example question using LCM. There are 18 dog cookies and 18 bones. So 3 packages of dog cookies and 2 bags of bones are needed. Dog cookies come in packages of 6, and bones in bags of 9. If there are 18 dogs, what is the least number of packages and bags needed so that each dog has a treat box with one bone and one cookie and there are no bones or cookies left over? Think of cookies in groups of 6. Think of bones in groups of 9. You are looking for the first multiple they share. Group Discussion They have given us the LCM of 18!!! How many of each item do we need?

Your Turn! Let’s Apply the Skill Linda is sending out invitations. If envelopes come in boxes of 25, and stamps come in packs of 10, what is the least number of stamps and envelopes she can buy to get one stamp for each envelope? 10: 10, 20, 30, 40, 50 25: 25, 50 Linda needs 50 stamps and envelopes. How many boxes of envelopes????2 How many packs of stamps????5 Listing the multiples to find the GCF works best here!!

Film TitleLength Introduction to the Museum2 minutes Profiles of Artists30 minutes Art and Architecture45 minutes Films play continuously at the museum. If the three films shown in the table above begin to play at the same time at 8:00 a.m., what time will it be before they begin playing together again? (Both methods are shown) We know the 2 is found inside the 30. All you need to use are the 30 and : 30, 60, 90 45: 45, 90 30: (3 x 10) 3 x 2 x 5 45: (9 x 5) X 2 x 5 x 3 LCM = 90 minutes The films will start again together in 90 minutes (1 ½ hours), or 9:30 a.m.

Did we meet the objectives? Did we find the LCM for sets of numbers? Did we use the LCM to answer word problems?