Ones Digit of 3^100

Date: 08/18/99 at 22:25:53
From: Stacy Payne
Subject: Exponents of 100

What is the number in the ones place of the value of 3 to the 100th 
power, and what is the reasoning?

I have found that the answer is 3486784401, but the reasoning has me 
stumped.

Date: 08/19/99 at 08:31:08
From: Doctor Rick
Subject: Re: Exponents of 100

Hi, Stacy.

I don't know what your answer means. The number in the ones place must 
be a digit, 0 to 9. If you worked out the whole answer for 3^100 (3 to 
the 100th power), you would get a number with 48 decimal places, so 
your answer isn't that, either.

Ah, I found it: your number is 3 raised to the 20th power. You stopped 
too soon - but don't bother to keep going.

My calculator can tell me what 3^100 is, but it only gives me the most 
significant digits, so I can't tell what digit is in the ones place.

There is a trick you can use to find the ones digit. I don't want to 
give it all away, but you can find it if you look for a pattern. Fill 
in this table:

     n   3^n
     -   ---
     0    1
     1    3
     2    9
     3    :
     4
     5
     6
     7
     8
     :

Once you see the pattern, you will be able to find the ones digit for 
any power of 3. But I hope you'll also want to try to understand the 
reason for the pattern; that's really the only way to be sure the 
pattern goes on forever.

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