LCM – with AlgebraStepping Up Our Level of Thinking

What does Y have to be?

Number Talk

Find the LCM of 4 and 6◦ List of Multiples for both 4 and 6.◦ 4: 4, 8, 12, 16, 20 . . .◦ 6: 6, 12, 18, 24, 30, . . .So the Least Common Multiple of 4 and 6 is 12Find the Least Common Multiple of 8 and 12List of Multiples8: 8, 16, 24, 32, 40, 4812: 12, 24, 36, 48, 60

First, let’s look at Least Common Multiples from a 6th grade perspective

You don’t need to make a whole list. The list can be done in your head.

Try finding the LCM of these◦5 and 12◦12 and 18◦10 and 15

Now that you’re in 8th grade

3660

30

Prime Factorization◦ Example: Find the LCM of 4 and 6◦ First, we could find the prime factorizations

4 = 2*2 6 = 2*3

◦ First you find the common factors, then you multiply that by the unshared factors

Here’s a different perspective on finding the LCM

Find the LCM of 12 and 15.◦ Prime Factorizations

12 = 2*2*3 15 = 3*5

Again, bring down the common factor(s), then multiply by the unshared factors.

Let’s look at another example

Using Prime Factorization, find the LCM of these◦ 20 and 30

◦ 8 and18

◦ 15 and 21

Try these for yourself

8: 2*2*218: 2*3*3Shared Factor is 2Unshared Factors are 2*2 and 3*3So 2*(2*2*3*3) = 72

20: 2*2*530: 2*3*5Shared Factors are 2 and 5Unshared Factors are 2 and 32*5 * (2*3) = 60

15: 3*521: 3*7Shared Factor is 3Unshared Factors are 5 and 7

So 3*(5*7) = 105

Look at the following prime factorizations written using exponents and the LCMs.

LCM of 20 and 30 was 60. 20 = 22 * 5 30 = 2*3*5

◦ 60 = 22*3*5

Using just the exponents, can you see how to find the LCM?

However, there was another way to find the LCM

LCM of 8 and 18 was 72 8 = 23

18 = 2*32

What is the largest exponent of the 2’s?◦23

What is the largest exponent of the 3’s?◦32

What does 23*32 =?◦72

Let’s look at another one

Find the LCM of 72 and 96◦ 72 = 23 * 32

◦ 96 = 25 * 3 What is the largest exponent of each prime

number?◦ 25 and 32

LCM = 25 * 32 =288

Let’s look at just one more.

What are the three ways we can find the LCM?◦ (6th grade) Make a small list of multiples and find the

least common multiple Good for small numbers

◦ (7th /8th grade) Use the prime factorization to locate shared factors and unshared factors. Multiply the single shared factors and multiply all the unshared factors. Works better for larger numbers

◦ (7th / 8th grade) Use the prime factorization and write them using exponents. Then multiply the values with the largest exponent for each prime number. Works better for larger numbers

Summary

Find the LCM of the following

15 and 35

12 and 21

96 and 50

200 and 49

Use any method you’d like. Some choices are wiser than others

Find the LCM of the following numbers 1) 15 and 55

2) 18 and 24

3) 25 and 12

4) 9 and 24

Warm-Up

Suppose I have xy and xz◦The factorization of xy and xz are xy: x*y xz : x*z

Let’s use the perspectives we learned earlier

xy: x*yxz: x*zShared Factor is xUnshared Factor are y and z

So the LCM is x*(y*z) = xyz

xy: x*y xz : x*z

What’s the largest exponent for x?◦ 1

What’s the largest exponent for y?◦ 1

What’s the largest exponent for z?◦ 1

LCM equals x*y*z = xyz

Alternatively

Factorization of both terms

Suppose I need the LCM of x2y3 and x2yz

x2y3: x*x*y*y*yx2yz: x*y*zShared Factors are x2 and yUnshared Factor are y2 and z

So the LCM is x2*y*(y2*z) = x2y3z

Using the alternative methodx2y3 X2yzWhat largest exponent for each variable?

x2 and y3 and z = x2y3z

Find the LCM of◦ a3b and ab2

◦ g5h2 and g2h

◦ O11f14 and O4f6

Try these on your own

a3b2

g5h2

o11f14

2x and y◦ 2x: 2*x◦ Y:y

Since they have no common factors, the LCM is the product of the two.

2x*y = 2xy 12x and 8x2

◦ 12x: 2*2*3*x = 22*3*x◦ 8x2: 2*2*2*x*x = 23*x2

LCM of 12 and 8 is 24 LCM of x and x2 is x2

LCM = 24x2

What about when we put coefficients in?

Find the LCM of ◦ 3x3 and 6x

◦ 10x2y2 and 15xyz2

◦ 3(x-4) and 9(x-3)

Try these on your own

6x3

30x2y2z2

9(x-4)(x-3)

Find the LCM of the following 1) xyz and z 2) 2x2y4 and 3x3y 3) (x-2) and (x-2)2

4) 9y and yz2

5) 5b2 and 15ab2

6) 2(m-1) and (m-3)

Warm-Up

It’s so we can add or subtract algebraic fractions◦ To add fractions that have algebraic expressions

in the denominator, we need a common denominator

Example

◦ The denominators need to be the same in order to add these. So we need the LCM of x and 3 first.

◦ LCM=3x

Why do we do this?

First, find the LCM of the denominators◦ LCM of 4y and x is 4xy.

◦ What do we need to multiply 4y by to make it 4xy?

◦ What do we need to multiply x by to get 4xy?◦ Put together◦ Then get your final answer

Some more fractions

Try these on your own

Since subtraction is ALMOST identical to addition. Try these next