Science:Math Exam Resources/Courses/MATH312/December 2008/Question 01 (d)
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Question 01 (d) 

For each of the following statements, indicate if it holds for every positive integers (if so, a simple true without a proof suffices, if not, a false together with a counterexample is expected). If , then . 
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 

There are two ways to approach this problem. First is trying all 10 possibilities for a and checking that only 4 works. The other involves isolating for 0, factoring and arguing what the possible answers must be. 
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 1 

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Please rate my easiness! It's quick and helps everyone guide their studies. The answer is false We try each combination for a
As we can see, we have two solutions given by and so the statement is false. 
Solution 2 

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Please rate my easiness! It's quick and helps everyone guide their studies. The answer is false. We isolate for 0 giving
Thus, we have that 10 divides and so we are looking for a value of a such that . We quickly see that works. This lifts to two solutions modulo 10, namely and this gives a contradiction. 