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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
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Re: how should i solve this question?

Quote:
 Originally Posted by alpah Find the last three digits of 9^105. Please show me how to solve this question. Thank you very much.
First, you could just calculate the number by multiplying by 9s sequentially as Infinity suggested (tongue-in-cheek). Actually, in this case that wouldn't be too bad -- 104 multiplications with up to 101 digits. But you can do better.

What you should realize is that the higher-valued 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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