User Name Remember Me? Password

 Number Theory Number Theory Math Forum

 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:
 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. Tags question, solve find the last three digits of 9^105

Click on a term to search for related topics.
 Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post tayfur Calculus 8 June 18th, 2013 09:42 AM kozhin Number Theory 1 February 17th, 2013 02:42 PM darkangel777 Math Events 1 January 27th, 2010 02:28 PM mungbean_jones Abstract Algebra 0 December 31st, 1969 04:00 PM darocker077 Abstract Algebra 0 December 31st, 1969 04:00 PM

 Contact - Home - Forums - Cryptocurrency Forum - Top      