Science:Math Exam Resources/Courses/MATH312/December 2008/Question 06 (c)
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Question 06 (c) 

The purpose of this problem is to prove the following tThe purpose of this problem is to prove the following theorem. Theorem 1. For all positive integers we have Let be positive integers. For , the theorem holds trivially, so we assume from now on that and write its primepower factorization as for different primes and positive integer exponents and some positive integer k. Let and focus on the prime power in the primepower factorization of m. (c) Prove that 
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Hint 

Split this problem up into the two cases as mentioned in parts (a) and (b). 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. We proceed in cases. Case 1: . In this case, part (a) states that . Since , the element above has an inverse showing that . Multiplying both sides by yields . which is what we wanted to show in this case. Case 2: . In part (b), we showed that
Since
We also have that
and hence that . 