Investigating Prime Factors as a Classroom Activity

Date: 06/27/2004 at 21:16:37
From: Gail
Subject: Teaching middle school mathematics

I'm a middle school math teacher, and I'm struggling with a question.
If n is a composite number, is it true that all prime factors of n 
must not exceed the square root of n?  I'm not even sure how to start
answering this question, or how to teach it to my class.

Date: 06/27/2004 at 23:03:29
From: Doctor Mitteldorf
Subject: Re: Teaching middle school mathematics

Dear Gail,

Don't be afraid that there are things you don't know.  The most
important thing you can impart to your students is how to approach a
new problem.  Think about this question, and model for them the way to
approach it.

Start with a few examples.  Pick a number--say 50.  What is the square 
root of 50?  About 7.  What are the prime factors of 50?  They are 2 
and 5.  Both are less than 7.  Ask your students to pick numbers, and 
to check what the prime factors are.  Ask if any of them can come up 
with any that have a prime factor greater than the square root of the 
number.
 
This in itself is a tremendous lesson for a 5th grade class.  They'll
learn a lot--about what a prime factor is, what a square root is, and
how you go about approaching a problem.  

Perhaps someone will notice that all factors come in pairs.  In other
words, if 5 is a factor of 50, then 50/5 is another factor.  Next,
maybe you'll notice that one factor of the pair is always greater than
the square root and one is less.  (Unless two are equal!)  Why is this?

If you're trying to come up with a counter-example (where there is a 
prime factor greater than the sqrt(n)) should you try a number with a 
lot of prime factors or just a few?  A large number or a small number?  
Remember that while proving a statement such as yours to be true takes 
some formal math, disproving it takes only one counter-example to show 
that it can be false.

Explore it and see what you find.  Now you're thinking like a 
mathematician!

- Doctor Mitteldorf, The Math Forum
  http://mathforum.org/dr.math/