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Learning Math Home
Number and Operation Session 6: Number Theory
 
Session 6 Part A Part B Homework
 
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Session 6 Materials:
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Session 6, Part A:
Models for Multiples and Factors

In This Part: The Venn Diagram Model | Finding Prime Factors | The Area Model

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. Note 4

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

Solution  

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

Solution  

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


 

Problem A4

Solution  

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


Next > Part A (Continued): The Area Model

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