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

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 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. (b) Prove that if then . You might want to do this as follows:

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Hint 

Again the question outlines the proof fairly thoroughly. 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. We prove the sub claims as outlines in the question. Prove that . For this, we use the property of the phi function to see
and thus by the multiplicativity of the phi function, we have
and hence . Prove that . The above shows that and this completes the proof. Prove that (here you may use without proof that .) We have that . Therefore, . We finish off this problem by noticing that . 