how you can tell from the prime factorization whether the least common
mutiple of two numbers is the product of the two numbers or is less
than the product of the two numbers?

Hello Joecliffe,

An easy to read explanation of this kind of work is at:
  http://www.purplemath.com/modules/lcm_gcf.htm
which describes a simple method to compute both the Least Common
Multiple (LCM) and the Greatest Common Factor (GCF). If you lay out
the factors as described on that page:
      2940 = 2 2 3   5   7 7
      3150 = 2   3 3 5 5 7
if there are no repeated values (vertically) between the two rows, the
LCM must be equal to the product of the two numbers. So in this
example, the LCM (44100) must be less than the product (9261000). [and
it is]

For another explanation - with a little more highlighting - that looks
at this in a similar matter is:
  http://www.bmcc.org/nish/MathTutorials/Numbers/n-lcm.htm
or looking at:
  http://www.ilovemaths.com/1lcmandhcf.htm
there is a nice explanation that the product of the two numbers =
LCM*GCF. So, if the GCF is 1, the LCM is the product of the two
numbers. (scroll down for this explanation) You can confirm using the
first example where:
  9261000 = 44100 * 210

Search phrases used include:
  least common multiple prime factors
  least common multiple prime factors "product of"

Please use a clarification request if this answer is unclear or you
need a more complete answer.
  --Maniac