Lesson: Problem Solving Principle
by batteredbutnotdefeated, Apr 6, 2010, 11:03 PM
6.
Show that the algebraic sum of any number of irreducible fractions whose denominators are all relatively prime to each other cannot be an integer.
In this proof, I will demonstrate the use of the problem solving principle "solve a smaller version of the problem":
solution
Show that the algebraic sum of any number of irreducible fractions whose denominators are all relatively prime to each other cannot be an integer.
In this proof, I will demonstrate the use of the problem solving principle "solve a smaller version of the problem":
solution
Since the problem at hand seems somewhat hard to cope with, we will approach a similar but smaller problem. Let's take the case for
:

For the previous equation to be an integer, there must be some number
such that:

However, for this to be true,
must equal some multiple of
and
must equal some multiple of
, which is clearly a contradiction. It's easy to see that this argument can be generalized for
.


For the previous equation to be an integer, there must be some number


However, for this to be true,




