Number and Operations Session 5, Part C:Factors

Session 5, Part C:
Factors (35 minutes)

You saw that you could devise tests for such numbers as 6 and 18 based on their relatively prime factors. Let's explore factors further.

A prime number is a number with exactly two factors. For example, the number 1 is not a prime number because it only has one factor, 1. The number 3 is a prime number because it has exactly two factors, 1 and 3. Note 4

Some numbers factor into two factors only, while others may have two factors, one or both of which can be factored further. Note 5

An important distinction can be made between the terms "factor" and "prime factor." By factors, we mean all the factors of a number. To find all the factors of 12, you can list them as shown below:

1 • 12
2 • 6
3 • 4

You know you can stop here because the next factor on the left would be 4, and you already have it listed on the right.

To find the prime factors of 12, you could use a factor tree:

The numbers on the bottom branch of this tree are the prime factors of 12 -- they can't be factored any further. So we say that 12 has only two prime factors, 2 and 3, and the prime factorization of 12 is 2² • 3. Note that we could have started the factor tree with the factors 3 and 4, and we would have derived the same prime factorization, 2² • 3.

Use the Interactive Activity to explore the numbers from 2 to 36. Which of these are prime? Which have only two factors? Which can be factored into more than two factors?

Drag each number sequentially into its appropriate column on the grid. When you drag a number into the correct column, you'll see its factors. Watch for any patterns that emerge.

This activity requires the Flash plug-in, which you can download for free from Macromedia's Web site. There is a non-interactive version of this activity for those who do not have access to the Flash plug-in.

The fundamental theorem of arithmetic states that every positive integer other than 1 has a unique factorization into primes (up to rearrangement of the factors). Now, any negative integer is simply 1 times a positive integer. So we can extend the theorem to all integers in a natural way: Each integer (except, of course 1, 0, and 1) can be written uniquely as a product of primes and either +1 or 1.

Here are some unique prime factorizations:

12 = 1 • 2² • 3
(Note that we usually leave off the 1 for positive numbers.)

-12 = -1 • 2² • 3

Problem C1

Use the Interactive Activity to explore the factors and prime factorizations of the numbers from 2 to 36.

a.	What are the factors of the numbers from 2 to 36?
b.	What are the prime factors of the numbers from 2 to 36?

Problem C2
Look at the prime factorizations of numbers. Do you see any patterns -- for example, how many factors in total a number will have based on its prime factorization?

Make a table that allows you to complete the prime factorization and total number of factors. Using numbers that are powers of 2 may be a useful way to see that pattern initially. Close Tip

Next > Homework

Session 5: Index | Notes | Solutions | Video