I introduced modular arithmetic in an earlier post (Introducing Modular Arithmetic via Group Theory). Let us look an application of this theory in the following problem.
Problem: Prove that there are no perfect squares ending in 99.
On the surface, this seems incredibly hopeless. After all, there are infinitely many number like this: 99, 199, 299, 399, etc. The first few certainly are not perfect squares but how can one guarantee that there are never any. This is where modular arithmetic comes to the rescue.
We will break up our infinite set of integers into four classes. Call them 
, 
, 
and 
depending on its remainder on division by 4. Note that any number ending in 99 belongs to 
because any such number can be represented as 
which leaves a remainder of 3 when divided by 4. Thus, our job is to show that this class has no perfect squares.
Well, each perfect square is either the square of an odd integer or of an even integer. In the first case, the perfect square has the form 
. Thus, such an integer belongs to . In the latter case, the perfect square has the form 
. Thus, such an integer belongs to . This means that all perfect squares belong to or . Hence, any number ending in 99, which is in , cannot be a perfect square.
Comments are closed