Science:Math Exam Resources/Courses/MATH437/December 2011/Question 01
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Question 01 

Let n be an integer such that . Prove that . 
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 

Prove this by contradiction. 
Hint 2 

A helpful idea is to consider the smallest prime dividing n. 
Hint 3 

Combining Fermat's Little Theorem as well as notice that if
and
then
will help complete the problem. 
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. Assume towards a contradiction that . Let p be the smallest prime dividing n. Notice that is odd and so p is not 2. Thus, we have that
and by Fermat's Little Theorem, we also have that . Thus, we also have that (via the Euclidean Algorithm) . However since p was the smallest prime dividing n. This gives us that
which is a contradiction. 