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

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. (d) Complete the proof of Theorem 1 above 
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 hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! 
Hint 

Do parts (a) to (c) for each prime factor. 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. Notice that parts (a) to (c) held for an arbitrary prime factor of m. We repeat the above for each such prime factor. Since they are all distinct primes, the fact that
holds for each will together imply that via the Chinese Remainder Theorem
and this completes the proof. 