Science:Math Exam Resources/Courses/MATH312/December 2008/Question 02 (a) iv
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Question 02 (a) iv 

For each of the following congruences, find the least nonnegative integer x that satisfies it.

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 

Reduce the right hand side using Euler's Theoerm (or Fermat's Little Theorem). Then isolate for x. 
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. We first reduce the right hand side using Euler's Formula (or Fermat's Little Theorem). As and
and hence
Now, we have
and thus, multiplying both sides by the inverse of 9 gives
(notice that we didn't need to compute this inverse, though if we wanted to the inverse is 2). 