Science:Math Exam Resources/Courses/MATH312/December 2009/Question 03 (a)
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Question 03 (a) 

Find the last three decimal digits of the number . 
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint. 
Hint 1 

It suffices to consider the number modulo 1000. 
Hint 2 

Euler's theorem will help. 
Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. Since considering a number modulo 1000 gives the last 3 digits, it suffices to consider this number modulo 1000. Now
Euler's Theorem (valid throughout since ) states that
and so we have
Let's examine . Notice that
and so by Euler's Theorem, we have . Thus for some integer s. Further,
and so giving for some integer t. Thus, . So . Hence . To complete the proof, we need to find the inverse of 7 modulo 1000. We can do this via the Euclidean algorithm.
and back substituting gives
and so the inverse of 7 modulo 1000 is 143. Hence
completing the question. 