Science:Math Exam Resources/Courses/MATH312/December 2005/Question 11
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Question 11 

Prove In this section, prove the statement given to you. Suppose a and N are integers with . Show 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 hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! 
Hint 

This can be solved either by induction or using the binomial theorem. 
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 1 

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Please rate my easiness! It's quick and helps everyone guide their studies. We proceed by induction. The case is easy. For , we have
Suppose the claim holds for N. Then for , we have
and so by mathematical induction we are done. 
Solution 2 

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Please rate my easiness! It's quick and helps everyone guide their studies. We can use the binomial theorem to see that
where all the terms in the right most summation are divisible by . 