Science:Math Exam Resources/Courses/MATH312/December 2010/Question 01
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Question 01 

Let n be an integer. Show that there does not exist a prime number which divides both 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 hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint. 
Hint 1 

Proceed by contradiction. Write out what it means for p to divide the given numbers. 
Hint 2 

Eliminate for n and factor to reach a contradiction. 
Checking a solution serves two purposes: helping you if, after having used all the hints, 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. Proceed by contradiction. Suppose that there exist integers x and y such that and . Multiplying the first by 5 and the second equation by 8 gives
subtracting the two equations gives and as p is prime, it divides the left hand side but not the right hand side which is a contradiction. Hence no such prime number can exist. 