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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


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


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 
help you some more.

- Doctor Jaffee, The Math Forum

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

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

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

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