Finding the Units Digit

Date: 02/26/2003 at 20:58:18
From: Mrs.Dremy
Subject: Exponents

My daughter got a math problem stating to find the ones digit of 
2003 to the 2003 power.

We do not know how to solve this problem and the calculator won't work.

Date: 02/26/2003 at 23:16:14
From: Doctor Peterson
Subject: Re: Exponents

Hi, Mrs. Dremy.

This kind of problem is meant to make the student think in ways a 
calculator can't. Try thinking about where the units digit of the 
answer will come from, as we start multiplying:

      2003
    * 2003
    ------
      6009
     6009
    ------
     66099

The only thing that affects the units digit of the product is the 
units digits of the factors. The 9 in the answer comes only from the 
3*3 in the factors we multiplied. That will continue to be true as we 
continue multiplying. So we see that the answer will be the same as 
if the question had been

    What is the units digit of 3^2003

That helps a bit, but we still don't want to do 2002 multiplications. 
So we can start multiplying, but keep an eye out for anything more 
that might help:

    3 * 3 = 9
    3 * 9 = 27
    3 * 27 = 81

Ah. Again, we don't need to worry about anything but the units digit, 
so we can continue with just the 7 from the 27, and we'll still get 
the right units digit:

    3 * 7 = 21

Again, just use the 1:

    3 * 1 = 3

Now we've come back to where we started. If you keep multiplying, 
you'll see that everything repeats. Putting together what we've found, 
we see that

    3^1 = 3
    3^2 = 9
    3^3 = x7
    3^4 = x1
    3^5 = x3
    ...

where the "x" means we don't care about any digits on the left.

You can use this pattern to find the answer to your problem.

If you have any further questions, feel free to write back.

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