Topic: Prime Factorization
Introduction:
The Fundamental Theorem of Arithmetic states that every number is either a prime
number or can be expressed as a unique product of primes. Example: 22 = 11 * 2
(which are both prime numbers).
A Prime Number is a number that can only be divided by itself and 1. That means it
has no other factors (or divisors). The first ten prime numbers are listed: 2, 3, 5, 7, 11,
13, 17, 19, 23, & 29
Euclid proved that there cannot be a largest prime number. The search is still on for
the current largest prime.
Activity: Using the Internet, Research the current search for the largest prime number.
Topics to include in your research:
· Dates that large primes were discovered?
· Techniques utilized?
· Where were these mathematicians located?
· What are the future directions for research in this area of mathematics?
Prime or Not?
As with many topics in Mathematics there are a number of rules used to determine
whether a number is prime or not. These rules are listed here:
A Natural Number is Divisible by:
2 If the Number is Even
3 If the Sum of the Digits is divisible by 3
4 If the last 2 digits is a number divisible by 4
5 If the last digit is 5 or 0
6 If the Number is even and divisible by 3
8 If the number is divisible by 4 and the result is even
9 If the sum of its digits is divisible by 9
10 If the last digit is 0
Activity: Using the rules above are the following numbers prime? List the first divisor
that the number is divisible by and list the rule used.
1. 14
2. 7
3. 1239834567
4. 693
5. 410
Factoring using the Tree Method:
A common method for factoring a number down to its prime factors utilizes the “Tree
Method”. This method is illustrated in the following graphic.
·
·
·
·
In this example we take 27 and notice that 2 + 7 = 9 which we know is divisible by 3.
We then notice that the first factors are 3 * 9 = 27.
Then we notice that 9 is further divisible by 3. In fact, 3 * 3 = 9.
Now we know that 3 is a prime number so we have in fact factored 27 down to its
prime factors: 3, 3, and 3.
This can also be shown using a graphing calculator program located at
www.mste.uiuc.edu/dildine/times/primfac2.8xp
Activity: Using the “Tree Method” factor the following numbers into their prime
factors and check them using the calculator program or the Internet tool located here:
www.mste.uiuc.edu/dildine/times/prime.html
1. 210
2. 453
3. 231
4. 36
5. 81
6. 72
Resources for additional exploration:
MSTE Grad Students have developed a few resources to explore prime numbers:
www.mste.uiuc.edu/activity/prime/prime.html presents a Java Applet designed to choose
random integers and check whether they are relatively prime. It was written by Nick
Exner.
Jim Dildine has also developed a simple script to determine if a number is prime (up to
16 digits long). It is located at:
www.mste.uiuc.edu/dildine/times/prime.html
Jim has also written a TI – 83, 83+, 84+ program that will factor a number down to its
prime factors.
www.mste.uiuc.edu/dildine/times/primfac2.8xp