I asked students to look at the number of factors for prime numbers, square numbers, and composite numbers made from two primes. This resulted in students uncovering the following:
- Prime numbers always have two factors
- Squares of prime numbers have 3 factors
- Composite numbers made from two prime numbers have 4 factors.
I then focused on the number 6 and asked why it would have 4 factors. I asked the class to consider this from a counting perspective. Students still were unsure, so I asked them to look at the individual factors for the two prime numbers (we knew that 2 had 2 factors and 3 had 2 factors). The factors of 2 are 1 and 2 while the factors of 3 are 1 and 3. How many ways can these factors be combined?
I used a tree diagram to help illustrate what was happening. Some students started to have a glimmer of understanding. I next drew a tree diagram for 30. It started off exactly like the one for 6. We then had to include two new branches off of each end. This results in 8 total factors for 30.
I then turned to 120 and wrote down its prime factorization: 23 x 31 x 51. I wrote down that 120 possessed 16 factors. One student wanted to know if I had memorized that value. I replied that I was making use of what we were doing to calculate the number of factors. We talked about how 23 would generate 4 factors. I drew a tree diagram to illustrate the point. I asked the class what would happen when we added the factors of 3 onto the diagram. The result would double. When we add the factors of 5 onto the diagram the result doubles again.
You can quickly count the number of factors an integer has by adding one to each exponent and then multiplying the values together. For 120, it has (3+1)(1+1)(1+1) = 16 total factors.
I asked the class to determine how many factors 1872 has for homework.
Visit the class summary for a student's perspective and to view the lesson slides.
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