Square Numbers End in 0 1 4 5 6 9

Date: 10/15/2002 at 13:27:31
From: Holly Gray
Subject: Square numbers

Why do all square numbers end in 0 1 4 5 6 9?

Yours sincerely, 
Holly

Date: 10/15/2002 at 14:13:31
From: Doctor Ian
Subject: Re: Square numbers

Hi Holly,

Here's how I would think about it. Suppose I have a number like 2234, 
and I want to find the square. I can multiply the number by itself, 
     
    11  
    11          <-- carries

   2234
 * 2234
 ------
   8936
  6702
  ...
  -----
   ...6

Now, note that what ends up in the ones column depends _only_ on the 
digits in the ones column.  None of the other digits can possibly 
affect it. So all of the following will have exactly the same thing in 
the ones column:

    4 *    4 =   16
   14 *   14 = ...6
  214 *  214 = ...6
 5214 * 5214 = ...6

and so on. So all we have to do is look at what happens when we square 
single digits:

  0 * 0 =  0   ends in 0
  1 * 1 =  1           1
  2 * 2 =  4           4
  3 * 3 =  9           9
  4 * 4 = 16           6
  5 * 5 = 25           5
  6 * 6 = 36           6
  7 * 7 = 49           9
  8 * 8 = 64           4
  9 * 9 = 81           1

So, unless we invent some new digits, there's no way to get a square 
to end in anything but 0, 1, 4, 5, 6, or 9.  

Does this make sense?

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