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

MATH312 December 2008

  •  Q1 (a)  •  Q1 (b)  •  Q1 (c)  •  Q1 (d)  •  Q1 (e)  •  Q1 (f)  •  Q2 (a) i  •  Q2 (a) ii  •  Q2 (a) iii  •  Q2 (a) iv  •  Q2 (b)  •  Q3 (a)  •  Q3 (b)  •  Q3 (c)  •  Q4 (a)  •  Q4 (b)  •  Q4 (c)  •  Q5 (a)  •  Q5 (b)  •  Q5 (c)  •  Q6 (a)  •  Q6 (b)  •  Q6 (c)  •  Q6 (d)  •  

   Other MATH312 Exams

  •  December 2010  •  December 2008  •  December 2005  •  December 2013  •  December 2012  •  December 2009  •  

Question 01 (e)
For each of the following statements, indicate if it holds for every positive integers $\displaystyle a,b,c$ (if so, a simple true without a proof suffices, if not, a false together with a counterexample is expected). If $\displaystyle \gcd(a,b)=1$, then $\displaystyle a^{b-1}\equiv 1\mod {b}$.

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
This should remind you of a mixture of Fermat's Little Theorem and Euler's Theorem. Can you use these to guide your solution?

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. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. The statement is false We seek a counter example and we take the smallest composite number for b, namely $\displaystyle b=4$. We then take $\displaystyle a=3$. Then $\displaystyle 3^{4-1}\equiv 3^{3}\equiv 27\equiv 3\not \equiv 1\mod {4}$. For a more enlightening proof, recall that by Euler's Theorem $\displaystyle a^{\phi (b)}\equiv 1\mod {b}$ Thus if we find a value of b where $\displaystyle \gcd(\phi (b),b-1)=1$ we would be able to construct a counter example since in this case there are integers s and t such that $\displaystyle s\phi (b)+(b-1)t=1$ and so choosing an a coprime to b between 2 and b-1 would give us that $\displaystyle a\equiv a^{s\phi (b)+(b-1)t}=(a^{\phi (b)})^{s}(a^{b-1})^{t}\equiv 1\mod {b}$ which would be a contradiction. This is what we did above.

Click here for similar questions

MER QGH flag, MER QGQ flag, MER QGS flag, MER RT flag, MER Tag Euler's theorem, MER Tag Fermat's little theorem, Pages using DynamicPageList3 parser function, Pages using DynamicPageList3 parser tag