MATH312 December 2008
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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 prime-power factorization as for different primes and positive integer exponents and some positive integer k. Let and focus on the prime power in the prime-power factorization of m.
(c) Prove that
Hint: Combine the results of (a) and (b).
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?
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Split this problem up into the two cases as mentioned in parts (a) and (b).
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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
We also have that
and hence that
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