Analyzing the Results of Large Exponents

Date: 09/10/2007 at 22:56:52
From: Katie
Subject: 1998^1998

What is 1998^1998 to the units digit?  How do you solve these kind of
problems?  Too big a number!

Date: 09/11/2007 at 11:13:10
From: Doctor Ian
Subject: Re: 1998^1998

Hi Katie,

A good place to start is by finding 1998^2:

    1998
  x 1998
  ------
    1998 *    8           15984
    1998 *   90   ->     179820
    1998 *  900         1798200
  + 1998 * 1000       + 1998000
  -------------       ---------
                        ??????4

Now, the interesting thing here is that the units digit of the product
is affected ONLY by the units digits of the factors.  So 1998*1998
must have the same units digit as just 8*8.

Convince yourself that this is true before continuing.  

It would follow, then, that 1998^3 and 8^3 should have the same units
digit, right?  Let's check:

  1998^3 = 7976023992

     8^3 =        512

So really, you have a much simpler question to answer:

  The units digit of 8^1998 is what? 

That's still a big exponent, so you still need to be clever to avoid
just evaluating the expression.  You can do that by looking at powers
of 8, and seeing if you can find a pattern in the units digit:

  8^1 =        8
  8^2 =       64
  8^3 =      512
  8^4 =     4096
  8^5 =    32768
  8^6 =   262144
  8^7 =  2097152 

and so on.  
   
Is this enough to get started?  Let me know if you need more help.

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

Date: 09/11/2007 at 20:43:09
From: Katie
Subject: Thank you (1998^1998)

Thank you very much for answering my question.  You really helped me
to understand that.  Thanks a lot!