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### 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

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/
```
Associated Topics:
Middle School Exponents

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