Math 7 ( Period 4)

LEAST COMMON MULTIPLE  5.3

Mrs. Lovetoteach had her kindergarten class sorting a pile of buttons. When they separated the pile into groups of 5, there were 3 leftover. When they separated them into groups of 7, there were 3 leftover. When they separated the pile into groups of 9, there were NONE leftover. what is the LEAST number of buttons the kindergarteners could have had?  Hint; you may want to set up a guess and check chart or draw pictures of what’s happening, or use another strategy that you think might work?

You’re taking care of your next door neighbor’s house for the next five week while they are away. These are your chores
1) Take the mail and newspaper in each day

2) Feed the fish every 3 days

3) Take out the garbage every 6 days

4) Feed the snake every 4 days

What is the first day that you will be doing all 4 chores on the same day? How often will you be doing all 4 chores on the same day during those 5 weeks?

This question CANNOT be answered using the GCF

You are NOT trying to find a factor of the days—but instead a Multiple

How to find the LCM  smallest number that your numbers can go into.

Just like the GCF let’s look at the letters backwards to understand it

MULTIPLE  each number given in the problem must go into this number multiples of 2: 2, 34, 6, 8,…

Multiples of 3: 3, 6, 9, 12…

Multiples of 5: 5, 10, 15, 20…

COMMON—must be a number that ALL of the numbers can go into

LEAST  must be the SMALLEST number that all of the numbers go into

There are the same ways to find it as the GCF

1) List all the multiples of each number and circle the smallest one that is common to all the numbers (Yuk- way too much work, if you ask me)

2) Circle every factor in the prime factorization of each number that is different and multiply

3) List the EXPANDED FORM prime factorization in a table and bring down ONE of EACH COLUMN. Then multiply (or you can do this with exponential form- you need the HIGHEST power of each column.

WINDOW BOX Method- My favorite

Create the window BOX method as if you are finding the GCF. The GCF will be on the left as usual.  But your LCM makes a capital L – find the product of the GCF and the last row of your window box ( the last row must be relatively prime) . You can also  multiply one of your numbers with the bottom relatively prime number of the opposite number.

The difference between GCF and LCM

For the GCF you need the LEAST POWER of only the COMMON FACTORS.

For the LCM you need the product of  GREATEST POWER of EVERY FACTOR

Why do we need EVERY FACTOR this time?

because it is a multiple of all your numbers

Mutiples start with each number so all the factors that make up each number have to be in the common multiple of the numbers.

Finding the LCM of 12 and 15. That multiple must be a multiple of 12

12: 12, 24, 36, …AS WELL AS

15: 15, 30, 45, ….

So the COMMON multiple just include

12: 2  x 2 x 3

and 15: 3 x 5

The LCM must have two 2’s and one 3 or 12 won’t do into it. It must also have that same 3 that 12 needs and one 5 or it won’t be a multiple of 15

Find the LCM 54 and 36

You could list all the multiples

This is more difficult than listing method for GCF for 2 reason

You don’t know where to stop as you list the first number—multiples go on forever…

the numbers get big very fast because they are multiples—not factors

Prime Factorization

54 = 2x3 x3 x3

36= 2 x 2 x 3 x 3

The LCM will be one of each factor of each number. (don’t double count a factor that is common to both numbers.

LCM = 2 x 2 x 3 x 3 x 3 = 108

What if by mistake you double up on a factor and use all of them? You will still get a common multiple—it won’t be the least common multiple. Generally you will get a really big number and the bigger the number is, the harder it is to use.

BOX Method

Checking the LCM to make sure it works

To show that the LCM works or each number, use the fraction simplifying concept. The LCM is on the top and the numbers are on the bottom each time

LCM =108

The numbers were 36 and 54

108/36 = 3

and

108/54 = 2

Use this same approach to prove the GCF works but this time you would put the GCF in the Denominator because it needs to go into the numbers

The numbers are still 36 and 54

We found that the GCF was 18

36/18 = 2

54/18 = 3