Science:Math Exam Resources/Courses/MATH312/December 2012/Question 05
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Question 05 

Show that if a and b are relatively prime integers, then . What is the multiplicative inverse of modulo 16? 
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 

For the first part, use Euler's Theorem. For the second part, simply evaluate it. 
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 

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Please rate my easiness! It's quick and helps everyone guide their studies. Euler's Theorem states that
Translating the above, there exist integers s and t such that
Taking the products of the left hand sides of the equation and the right hand sides of the equation yields (below we have flipped the sides of the equations)
As and are at least one, we have that
and hence we get the required result. For the last part, we have
(or easier, notice that so Euler's theorem gives the same result) Thus as the number is 1, its inverse it itself. 