MATH312 December 2009
• Q1 (a) • Q1 (b) • Q2 (a) • Q2 (b) • Q3 (a) • Q3 (b) • Q4 (a) • Q4 (b) • Q5 (a) • Q5 (b) • Q6 (a) • Q6 (b) •
Question 01 (b)
Suppose that a and b are integers with . Then show that is either 1 or 2.
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!
Perform the Euclidean algorithm by using long division.
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Using long division, we see that
and thus, the Euclidean algorithm states that
Now, as a and b are coprime, we see that
. If this were not true then some factor of b would divide and that means that this factor would also divide a which means that the factor must have been 1 since a and b were coprime. Finally, it's clear that is 1 or 2. This completes the proof.
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MER QGH flag, MER QGQ flag, MER QGS flag, MER RT flag, MER Tag Euclidean algorithm, Pages using DynamicPageList parser function, Pages using DynamicPageList parser tag