Number Theory Part A: Models for Multiples and Factors (75 minutes)

In This Part: The Venn Diagram Model
The numbers 24 and 36 have certain things in common, including many common factors — numbers that divide evenly into both of them. For example 2, 3, and 6 are all common factors. The largest such number is called the “greatest common factor.” In this case, the greatest common factor of 24 and 36 is 12. No number greater than 12 is a factor of each of these numbers. See Note 2 below.

Another characteristic numbers can share is a common multiple — a third number that is evenly divisible by both 24 and 36. The smallest such number is called the “least common multiple.” In this case, the least common multiple is 72. No number less than 72 is evenly divisible by each number.

One way to explore the common factors and multiples of the two numbers is to use a Venn diagram:

The circle on the left contains all the prime factors (i.e., counting numbers that have exactly two factors: themselves and 1) of 24, and the circle on the right contains all the prime factors of 36. (The number 1 doesn’t qualify as prime, because it has only one factor.)

The numbers contained in the intersection are those factors that are in both numbers; i.e., their common factors. That means that the 2s and the 3 in the intersection, both separately and multiplied together (2 • 2, 2 • 3, and 2 • 2 • 3 — or 2, 3, 4, 6, and 12), are all common factors.

Note that the largest of these factors is 12. The greatest common factor (GCF) of 24 and 36 is 2 • 2 • 3, or 12, the product of all the numbers in the overlap.

Since the circle on the left contains all the factors of 24, every multiple of 24 must contain all of these factors. Likewise, since the circle on the right contains all the factors of 36, every multiple of 36 must contain all of these factors.

In order to be a multiple of both numbers, a number must contain all the factors of both numbers. The smallest number to do this is 2 • 2 • 2 • 3 • 3, or 72, the product of all the factors in the circles. Thus, the least common multiple (LCM) of 24 and 36 is 72.

See Note 3 below.

Problem A1
Use a Venn diagram to determine the GCF and LCM for 18 and 30.

In This Part: Finding Prime Factors
The numbers we’ve examined so far have been fairly simple to factor. Now let’s look at a general method for finding prime factors. See Note 4 below.

One method is to draw a factor tree. To do this, write a number — 24, for example — and then draw an upside-down V under it. This V represents two “branches” of the factor tree. Think of a pair of numbers with the product 24; for example, 4 and 6. Check to see if either of these numbers is prime. In this case, the answer is no.

Draw another V under each number that is not prime, and find two factors for each of these numbers. In this case, we will find the factors 2 and 2 for 4, and 2 and 3 for 6. Now we have four factors, 2, 2, 2, and 3, all of which are prime numbers. This is the prime factorization of 24.

In mathematics, we like to do things via consistent algorithms. So rather than just picking two factors, it’s a good idea to make the process more consistent by first finding the smallest prime factor and its partner and then repeating that process on the partner (since the first number is guaranteed to be prime). For example, you could factor out all of the possible 2s, then all the 3s, then 5s, 7s, and so on, until the number is completely factored. Here is what the diagrams look like for the numbers 24 and 36:

Problem A2
Does the order in which you factor a number matter? Is the product always uniquely that one number? To answer these questions, use a factor tree to find the prime factorization of 60 in the following ways:
a. Start by factoring out 2s.
b. Do another diagram, but this time start by factoring out 10s.
c. Do a third diagram, but this time start by factoring out 6s.
d. What is the same and what is different about your results?

When you factor a number, no matter where you start, you always get the same set of factors; the only difference might be the order in which they occur. This phenomenon is called the fundamental theorem of arithmetic, which states that any integer (other than 0, and 1) can be factored into a product of prime numbers and that this product is unique except for the order of the factors. This is another reason why 1 cannot be considered prime — otherwise, this, and every other result that builds on it, falls apart.

For example, we could factor 6 in an infinite number of ways:

…and so on, for any number of 1s that we cared to use.

Problem A3
Draw a factor tree to find the factors of 231 and 195.

Problem A4
Use a Venn diagram to find the GCF and LCM of 231 and 195.

In This Part: The Area Model
The area model makes the process of finding GCFs and LCMs visual. See Note 5 below.

Greatest Common Factor
If we think of the numbers 24 and 36 as the dimensions of a rectangle, then it follows that any common factor could be the dimensions of a square that would tile that entire rectangle.

For example, a 1-by-1 square would tile the 24-by-36 rectangle without any gaps or overlaps. So would a 2-by-2 or a 3-by-3 square. Notice that these numbers are all common factors of 24 and 36.

To determine the GCF, we want to find the dimensions of the largest square that could tile the entire rectangle without gaps or overlap. Here’s one quick method.
Start with the 36-by-24 rectangle:

The largest square tile that fits inside this rectangle and is flush against one side is 24 by 24. Only one tile of this size will fit:

The largest square tile that fits inside the remaining rectangle and is flush against one side is 12 by 12. Two tiles of this size will fit. The original rectangle is now completely filled:

Least Common Multiple
Note that the 24-by-24 square could also be filled with the 12-by-12 tiles, so 12 by 12 is the largest tile that could fill the original 24-by-36 rectangle; therefore, 12 is the GCF of 24 and 36.

Conversely, if we think of 24 and 36 as the dimensions of a rectangle that could tile a square, then it follows that any common multiple could be the dimensions of a square that could be tiled by this rectangle.

For example, since 24 • 36 = 864, a square that is 864 by 864 could be tiled by the 24-by-36 rectangle. The LCM of 24 and 36 would be the dimensions of the smallest square that could be tiled by the 24-by-36 rectangle.

Here’s a quick method for determining the LCM.
Start with the 24-by-36 rectangle. Your goal is to make a square tiled with rectangles of these dimensions:

Since the width (24) is less than the height (36), add a column of tiles to the right of the rectangle (in this case, one tile). This makes a 48-by-36 rectangle:

The width (48) is now greater than the height (36), so add a row of tiles under the existing rectangle (in this case, two tiles). This makes a 48-by-72 rectangle:

The width (48) is now less than the height (72), so add another column (two tiles) to the right of the existing rectangles. The dimensions are now 72 by 72 — and you’ve made a square!

Use the following Interactive Activity to answer Problem A5. The 72-by-72 square is the smallest square that can be tiled with a 24-by-36 rectangle. Therefore, the LCM of 24 and 36 is 72.

For a non-interactive version of this activity, use graph paper when drawing the squares and rectangles you wish to represent to ensure that the dimensions of the shapes are precise.

Problem A5
Use the area model to find the GCF and LCM of the following:
a. 30 and 42
b. 18 and 30

Video Segment
In this video segment, Ben and Doug use the area model to find the GCF for two numbers, following the analogy of tiling a rectangle with the biggest square they can fit. Watch this segment after you’ve completed Problem A5.

Notice that the teachers omitted one step and didn’t use the square with the dimensions of 12-by-12 to tile the 12-by-30 rectangle. Think about why going through all the steps will ensure that the result will be the largest common factor rather than just any common factor.

You can find this segment on the session video approximately 6 minutes and 4 seconds after the Annenberg Media logo.

Problem A6
Can you explain in your own words why the area model works?

Video Segment
Here, Ben and Doug use the area model to find the LCM for two numbers, following the analogy of finding the biggest square that can be tiled with a rectangle whose dimensions are the two original numbers. Notice the connection they make between the area model and Venn diagrams.

You can find this segment on the session video approximately 8 minutes and 32 seconds after the Annenberg Media logo.