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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. See Note 4 below.
Some numbers factor into two factors only, while others may have two factors, one or both of which can be factored further. See Note 5 below.
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 2^{2} • 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, 2^{2} • 3.
Problem C1
This is the noninteractive version of Problem C1:
Complete the following tables to explore the factors and prime factorization of the numbers from 2 to 36.
a. What are the factors of the numbers from 2 to 36? Enter each number in the appropriate column in the table below. For example, enter 4: 1, 2, 4 in the Three Factors column and 16: 1, 2, 4, 8, 16 in the Five Factors column.
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 • 2^{2} • 3
(Note that we usually leave off the 1 for positive numbers.)
12 = 1 • 2^{2} • 3
36 = 3^{2} • 2^{2} will have (2 + 1) • (2 + 1), or 9 factors total.
TAKE IT FURTHER
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.
Note 4
Notice that, for our purposes, we’ve defined “factors” as a positive divisor, so prime numbers will also be positive.
Note 5
If you’re working in a group, you may want to play the Factor Game, which takes about 20 minutes. The Factor Game is a fun activity that requires understanding of factors; players must use some strategy to ensure a win!
Show participants a chart with the numbers 2 through 24:
2 – 24 



Explain that they will be divided into two teams, Squares and Circles. Each team in turn chooses one of the numbers. The other team then claims all of the numbers on the chart that are factors of that number (if they’re not already taken). When all the numbers have been chosen, the team with the highest sum wins the game.
For example, say the Squares team chooses 21 and draws a square around it. The Circles team then draws a circle around all of the factors of 21, in this case 3 and 7. Then the Circles team chooses 24 and circles it. The Squares team then puts a square around all its factors, 2, 4, 6, 8, and 12 (they can’t put a square around 3, because it’s already been taken). Teams alternate choosing numbers, continuing until all the numbers are taken.
There are many variations of this game. For example, you could start with more numbers on the board, or you could decide that teams are not allowed to choose numbers with no factors left on the board.
After playing the Factor Game, ask participants to describe a strategy that would help them win the next time they play. Would they rather be the first or second team to choose a number?
Problem C1
a.
b.
Problem C2
Looking at the prime factorization of numbers, you can tell how many factors a number will have in total. For example, the prime factorization of 2 is 2^{1}, and 2 has two factors in total, 1 and 2. The prime factorization of 4 is 2^{2}, and it has three factors in all: 1, 2, and 4. To further investigate this pattern, let’s look at the following:
36 = 3^{2} • 2^{2} will have (2 + 1) • (2 + 1), or nine factors total.
24 = 2^{3} • 3^{1} will have (3 + 1) • (1 + 1), or eight factors total.
81 = 3^{4} will have (4 + 1), or five factors total.