Science:Math Exam Resources/Courses/MATH312/December 2009/Question 02 (b)
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Question 02 (b) 

Determine whether 2821 is a Carmichael number. Explain your answer. 
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 

A Carmichael number is a composite number n which satisfies
for all integers with . 
Hint 2 

Recall Korselt's criterion which states that a positive composite integer n is a Carmichael number if and only if n is square free and for all prime divisors p of n, it is true that 
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. We follow the hints above (especially hint 2) which suggests we should use Korselt's Criterion. We first factor the number as
We check that for each prime factor. Notice that
We can see these are true by factoring . Hence 2821 is a Carmichael number. 