Math 6A ( Periods 1 & 2)

Least Common Multiple 5-6

Here is a review of that lesson...

Let’s look at the nonzero multiples of 8 and 12—listed in order

Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72…

Multiples of 12: 12, 24, 36, 48, 60, 72, ….

The numbers 24, 48, and 72, ... are called common multiples of 8 and 12. The least of these multiples is 24 and is therefore called the least common multiple.

LCM(8, 12) = 24

To find the LCM of two whole numbers, we can write out lists of multiples of the two numbers.

Or, we can use prime factorization

Lets find LCM(12, 15)

12 = 2²∙3

15 = 3∙5

The LCM will be made up of the greatest power of each factor

LCM will be 2²∙3∙5 = 60

The book has a third option or method

you can check out, if you’d like

Let’s find LCM (54, 60)

54= 2∙3∙3∙3 = 2∙3³

60 = 2∙2∙3∙5 = 2²∙3∙5

The greatest power of 2 that occurs in either prime factorization is 2²

The greatest power of 3 that occurs in either prime factorization is 3³

The greatest power of 5 that occurs in either prime factorization is 5

Therefore, LCM(54,60) is 2²∙3³∙5 = 540

REMEMBER:

The GCF (greatest common factor) is a factor. The GCF of two numbers will be either the smaller of the two or smaller than both

The LCM (least common multiple) is a multiple. The LCM of the two numbers will be the largest of the two or larger than both.

To find the LCM of two whole numbers you could write out the lists of multiples-- and that works relatively easily with small numbers... but there are more efficient ways to find the least common multiple of two whole numbers.

1. Write out the first few multiples of the larger of the two numbers and test each multiple for divisibility by the smaller number. The first multiple of the larger number that is divisible by the smaller number is the LCM

2. You can use prime factorization to find the LCM. The LCM is the PRODUCT OF EVERY factor to its GREATEST power!!

LCM(54, 60)

54 = 2⋅ 3⋅ 3⋅ 3 = 2⋅ 3³

60 = 2⋅ 2⋅ 3⋅ 5 = 2²⋅ 3⋅ 5

So the greatest power of 2 is 2²

The greatest power of 3 is just 3

and the greatest power of 5 is just 5

so the product of 2²⋅ 3⋅ 5 will be the LCM

LCM(54, 60) = 540

Second Day:

3. You may use the BOX method as shown in class... unfortunately it does not show well here. Remember you need to create a L The numbers on the side of the box represent the GCF!! You need to multiple them with the last row of factors.

See me before or after class if you want any review!!

We reviewed the concept of relatively prime and noticed that any two prime numbers are relatively prime. We also noticed that if two numbers are relatively prime-- neither of them must be prime....

We also found out that if one number is a factor of a second number, the GCF of the two numbers is the first number AND... if one whole number is a factor of a second whole number the LCM of the two numbers is the second number!!

GCF(12,24) = 12

LCM(12,24) = 24

WOW!!

If two whole numbers are relatively prime---

their GCF = 1

and their LCM is their product!!

GCF(8,9) =1

GCF(8,9) = 72

WOW!!

LCM & GCF Story Problems

1) Read the problem

2) Re-read the problem!!

3) Figure out what is being asked for!!

4) find the "magic " word... to help you determine if you are finding GCF or LCM

5) When in doubt... draw it out!!