Science:Math Exam Resources/Courses/MATH437/December 2006/Question 01
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Question 01 

Suppose that x,y,z are positive integers satisfying and . Show that there are integers u and v for which and . 
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 

The Fundamental Theorem of Arithmetic is the major idea in this proof. Start with the term and proceed from there. 
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. By the fundamental theorem of arithmetic, let be the prime factorization. Suppose that for any i. Then since x and y are coprime, this means that . Thus, we must have that . If one of x or y (or both) has no prime factors, then since these numbers are positive, they must be equal to 1 which is a square. Repeating this argument for all such i and noticing that this argument is symmetric in x and y, we see that
and
completing the proof. 