Math 10: GCF and LCM

Factor trees, factor trees everywhere

Also another review.

First we need a lesson on prime and composite numbers.
Prime numbers are numbers that are divisible by ONLY one and itself.
Composite numbers are divisible by one and itself, but also by two different factors.

Some prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, etc.

4 is a composite number because it has more than two factors: 1, 2, and 4.
12 is a composite number because it has six factors: 1, 2, 3, 4, 6, and 12.
*0 and 1 are not prime nor composite numbers.

Blah, blah, blah, divisibility rules, like even numbers are divisible by 2, blah. It's not even important anymore when some questions ask, "Is this divisible by 17? 13? 19? 101? Oh, by the way, you can't use a calculator."

Moving on.

Finding prime factors of composite numbers:
The book says three methods, but the second one is kinda useless.

Method 1: Factor Trees

2 x 2 x 3 x 3 = 36

 2 x 2 x 2 x 3 x 5 = 120

2 x 2 x 2 = 8

Method 2: Continuous method (or the "boot" method, but it doesn't look like a boot)

2 x 2 x 3 x 3 = 36

So now you know how to find prime factors. So what? Well, now we need to learn LCM and GCF so that you can find out when three people can work on the same day, how many months will pass for two planets to line up together again, and what each number has in common.

Terms to learn: 
GCF: Greatest common factor
LCM: Lowest common multiples


How to find the GCF of two numbers using a factor tree:

First make factor trees and fine the prime factors for both.

Find the common factors in both.

In this case, the GCF of 10 and 12 is 2.

How to find the LCM of two numbers using a factor tree:

Make the factor trees and find the prime factors.

Find out which number has more factors of the same kind and multiply them all together.

The LCM of 10 and 12 is 60.

How to find the GCF of two numbers using a boot:

Find the prime factors. If both numbers cannot be divided by the same number, bring the other number down.

Find the factor that did not involve a number that went down in that row.

The GCF of 10 and 12 is 2.

How to find the LCM of two numbers using a boot:
 

Find the factors again.

Multiply all the numbers outside the lines.

Again, the LCM is 60.

Done! Now you know how to find GCF and LCM!
BUT for you grade 10s out there, it's not over yet.

How do you solve this question?

 

Find factors.

Cross out the same numbers from both the denominator and numerator.

The simplified version of 140/728 is 5/26.

 

One last thing. Stay, it's a good one.
There is a trick in the book that shows you another way to find the LCM of two numbers.

Let's try to find the LCM of 10 and 12.:
1) Multiply the two numbers together. (10 x 12 = 120)
2) Divide that number by the GCF. (120 ÷ 2 = 60)

The LCM of 10 and 12 is 60.