MATH312 December 2008
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Question 06 (a)
The 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.
(a) Prove that if then
Hint: Use Euler's theorem and prove/use that
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Proceed as in the hint given in the problem. Notice that
holding since is multiplicative. Hence
and for simplicity, say for some integer s. Thus
where the last line holds by Euler's theorem since we are given in this question that so .
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