Greatest Common Divisor, and Least Common Multiple

Greatest Common Divisor is also known as the Greatest Common Factor (GCF). You may be given two numbers... let's say 18 and 24, and you are asked to find the GCF of both the numbers. "GCF (18,24) =" Our goal is to find the one factor that both of these numbers have that is the largest. This means that you want to find the one number that divides both of these into a number that has no remainder. There are a few numbers that divide them both, but we are interested in finding the number that is the greatest.

The first thing you need to do is to find the all the possible factors of each number:

18= {1,2,3,6,9,18}

24={1,2,3,4,6,8,12,24}

The factors they have in common are {1,2,3,6} because these are all of the numbers that divide them both without leaving any remainder, and we are looking for the number that is the greatest of them all, and we come up with the number 6 since 6x3=18 and 6x4=24. Therefore, GCF (18,24) = 6

Least Common Multiple

My favorite way to find the Least Common Multiple is a take off of the Factor Tree tool. Let's take a few numbers and find the LCM of them. LCM (18,24,12) One way of doing this could be taking each number, multiplying them by 1, then 2, then 3, etc. etc. until you find them all matching up and finding the LCM. However that is a bit time consuming, so using the Factor Tree to find the prime number factorization of the numbers, we'll find a quick and easy way to find the LCM of 18,24,12.

If we worked the factor tree we would find that the prime factorization would be as follows:

18=32x2

24=3x23 

12=3x22

To easily find the LCM of this set of numbers, we would first take the number 3 because all or one of the numbers has that number in the prime factorization. We would take that number 3 and square it, because that is the largest power of the number 3 that we see. Then we'll take the number 2 since all or one of the numbers contains the number 2. Since the largest one is 2 cubed we will use that. There are no other prime numbers used and we are going to take those numbers and multiply them to find the LCM:

32x23=72

(3x3x2x2x2=72)

This website explains factorization and finding LCM and GCD very well: purple math

Prime Numbers, Factor Trees, and Prime Factorization

Prime Numbers

One key that I have learned to really appreciate in finding factors is to know the prime numbers. A prime number is a number that cannot be divided among 2 numbers (excluding the number 1). 6 is not prime because it can be divided by 2 and 3. 2 cannot be broken down any more and neither can 3. The prime factors of the number 6 are 2 and 3. There are only a certain number of prime numbers and those are the ones we are looking for.

This chart will help for finding prime numbers quickly:

Video on polishing up your prime number skills...

Factor Trees and Prime Factorization

This is a wonderful tool that I have used over and over in factor finding, and have even found other uses for it just for fun. It's quick and easy to do on paper without a calculator if the numbers aren't to big, but if they get a little bigger you may want to use a calculator. The purpose of this exercise is to find the prime numbers of any given number by breaking in down piece by piece. My favorite way to do it is to see if the number can first be divided by 2. If not, then maybe 3, or 5.  Let's take the number 246...

246

/\

2  123

246 divided by 2 gives us 123..

2 cannot be broken down anymore because it's a prime number, but 123 can be..
 Lets keep going...

246
/\

2  123

    /\

     3 41

123 can be broken down to 3 divided by 41.
Since 41 cannot be broken down any more we consider it a prime number.
The prime factorization of the number 246 is 2x3x41.
This is not only a great tool just to have and to know how to do, but it will come in handy for other problem solving needs in the future, when you want to learn quick and easy ways how to find Least Common Multiples or Greatest Common Divisors. It's also just good to know the breakdown of any number, and enhances your math skills in general.

Click here for some math game ideas for learning factor trees in the classroom!

Base Ten vs Base Five

Let's stretch our minds a little! Let's take something that we take for granted every day in life, take it apart, mix it up, and see if it makes sense when we see it another light.

Counting! 

That is what most of us take for granted. We learn to count to 10 when we are 2 or 3 years old, and build on that for the rest of our lives. Learning to add and multiply, and work with all these numbers. Our place value depends on the number of 10, simply because of the fact that we have 10 fingers, but there are other cultures from different times that have had other bases and have learned to use numbers in completely different ways than we do, but with the same general concept. The reason we go through learning different bases is to remember how it is to learn how to think in brand new ways, just like it is for young kids who are learning math for the very first time.

Let's use this number:

1324

As we look at this, we're going to also remember our expanded notation. The 1 is not only a 1 but it represents 1,000 because it is in the thousands place. Just as the 3 represents 300 and 2 represents 20 and 4 simply represents 4. Each place value is an exponent of the number 10, which is our base. That is why it's called Base Ten. For example, 4 is 10 multiplied by ten to the zero power, 2 is multiplied by ten to the first power, 3 multiplied by ten squared, and 1 is multiplied by ten cubed. 

Now, what if we see this number and we are told it's in Base Five? That means, instead of multiplying each digit by exponents of Ten, we multiply them by exponents of Five! So it would look like this:

1324five = (1x125)+(3x25)+(2x5)+4

Notice how we are doing the same thing that we did with base 10. Except now that we are in base 5, all the numbers we multiply by are by exponents of 5. 1 times five cubed, 3 times five squared, 2 times five with no power and a simple 4. If we solve this problem we can figure out what this number translates to in Base Ten, which is our "native language" in math.

1324five = 125+75+10+4

So then, we end up with:
1324five = 214ten

A quick look at multiplying in other bases:

Here's extra help in converting to other bases

Expanded Notation

Expanded notation is a great tool that we can use to simplify math problems by breaking them down into bite size pieces in order to make more sense of the problem. The way we learned how to do expanded notation is in what we call "Base Ten". Basically, this means that each place value is some exponent of the number 10. The "one's" place is the number times ten to the zero power. The "ten's" place is that number times ten to the first power. The "hundred's" place is ten to the second power, or ten squared. The "thousand's" place is ten to the third power, or ten cubed, and so on.

Lets look at this number to write in expanded form:

7265

Instead of looking at this as one large and complex number, we can break this information down by recognizing place values. We will multiply each place value by the number 10's proper exponent for each place value. Starting with the 7 which will be multiplied by 10 cubed, moving on to the 2 which will be multiplied by 10 squared, and going to 6 which will be multiplied by 10 to the first power, and ending with 5 which will be multiplied by 10 to the zero power. Then we will add all the numbers together at the end. We break will break this number down and build it up to show place value. 

 7 is in the thousand's place 

2 is in the hundred's place

6 is in the ten's place

5 is in the one's place

So when we write this number in expanded notation it will look like this:

7265 = (7x1,000)+(2x100)+(6x10)+(5x1)

This skill is great for kids to learn in math, it helps them take something abstract and give it a foundation to build on. It also helps develop their thinking processes as they solve problems. When they begin to add, subtract, divide or multiply numbers, they can use this tool to solve problems easier and faster. A great tool to use when teaching the foundations of place value to kids is to introduce to them base ten manipulatives. They need to grasp the concept of place value in order to build on it later. Get them familiar with these base ten blocks in order for them to see how big is the number they're working with, which number belongs to which place value and what that really looks like as far as how many hundreds, tens, and ones there are.

Practice place value through a math game
Here you can see a video learning how to use expanded notation in multiplication: