Example 4

Proof Examples

 

Consider the following conjecture:

• If simplifies to an Integer and simplifies to an Integer, then also simplifies to an Integer.

1. Give two examples showing the conjecture works.
   
   Try substituting a = 2, b = 4, and c = 12.
   
   , , and
   
   Note that each expression simplifies to an Integer.
   
   Try substituting a = 3, b = 9, and c = 21.
   
   , , and
   
   
2. Prove the conjecture.
   
   To prove this conjecture, you can show that the general case works. Use variables to show the general case.
   
   You already know that simplifies to an Integer and simplifies to an Integer. Your goal is to show that also simplifies to an Integer.
   
   Using fraction rules you can write:
   
   
   
   Integers are positive or negative Whole Numbers or zero and adding two Integers will always give another Integer. As such, is also an Integer.