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### 3 to the Power of 1343

```Date: 07/22/2003 at 23:08:11
From: Gemma
Subject: Find the last two digits of 3 to the power of 1343

Find the last two digits of 3 to the power of 1343.

I have looked through your archives; however, I cannot find the
pattern for the 2nd to last digit of the multiples of three, i.e., the
pattern is

01,03,09,27,81,43,29,87,61,83,49,47,41,23,69,07,21,...

It is clear the last digit is 1,3,9,7 but how do I work out the 2nd to
last?

Thanks,
Gemma
```

```
Date: 07/23/2003 at 00:53:35
From: Doctor Jaffee
Subject: Re: Find the last two digits of 3 to the power of 1343

Hi Gemma,

Here is my suggestion. You already figured out that the last digit
must be 1,3,7, or 9.  Can you figure out which one?

Once you do, focus your attention on just those powers of 3 that end
in the digit you found. You will discover that the preceding digit
follows a very easy predictable pattern.

Give it a try and if you want to check your solution with me or if you
have difficulties or other questions write back to me and I'll try to

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

```
Date: 07/23/2003 at 01:12:43
From: Doctor Greenie
Subject: Re: Find the last two digits of 3 to the power of 1343

Hi, Gemma -

You stopped just too soon in your display of the pattern for the last
two digits. If you take the pattern a few more steps you will have
what you need to solve your problem.

You might have seen that the pattern was going to repeat soon by
looking at the patterns of next-to-last digits for each different last
digit. For example, for successive entries in your pattern with last
digit 1, the next-to-last digits are 0, 8, 6, 4, 2, ....

I think you can finish the problem on your own.

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

```
Date: 07/24/2003 at 05:26:49
From: Gemma
Subject: Thank you (Find the last two digits of 3 to the power of
1343)

Hey thanks! I got it now.
```
Associated Topics:
High School Number Theory

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