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December 25th, 2006, 09:27 AM  #1 
Newbie Joined: Nov 2006 Posts: 20 Thanks: 0  how should i solve this question?
Find the last three digits of 9^105. Please show me how to solve this question. Thank you very much. 
December 25th, 2006, 02:26 PM  #2 
Senior Member Joined: Dec 2006 Posts: 1,111 Thanks: 0 
Just get out a piece of paper and a pen and multiply 9 by itself 105 times, and then look at the last three digits when you're finally done with the calculation two or three years from now. Actually, I can tell you that the number will end in 9, but more than that, I don't know. The reason it will end is 9 is because any odd number of 9's multiplied together will result in a number ending in 9. ( Any even number of 9's multiplied together will result in a number ending in 1. ) 
December 25th, 2006, 09:51 PM  #3  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: how should i solve this question? Quote:
What you should realize is that the highervalued digits don't matter to you. The last three digits of 9^105 are the same as the last three digits of 1009^105, or 838958448983009^105 for that matter. This means that you could calculate the answer in 104 multiplications of only 3 digits each, a large improvement. But you can do much better yet! 9^105 = 9 * 9^104. Using the rules of exponents you (hopefully) know, you can see 9^104 = (9^2)^52 = (9^4)^26 = 6561^26. Of course you're only interested in the last three digits, so you can pretend that this is 561^26. Thus you know that the last three digits of your number is the same as the last three digits of 9 * 561^26. Continue this way to get an answer. If you post it here we can check it for you, if you'd like.  

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