Science:Math Exam Resources/Courses/MATH312/December 2010/Question 08

MATH312 December 2010

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Question 08
Prove that the congruence $\displaystyle x^{5}\equiv 1\mod {52579}$ has exactly one solution, $\displaystyle x\equiv 1\mod {52579}$. Use the fact that 52579 is a prime. What other property of 52579 is used in your proof?

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
Try see what Fermat's Little Theorem and the Euclidean Algorithm can do to attack this problem.

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. By Fermat's Little Theorem, we have that $\displaystyle x^{52578}\equiv 1\mod {52579}$. The given problem tells us that $\displaystyle x^{5}\equiv 1\mod {52579}$ Using the Euclidean Algorithm, we can find integers a and b such that $\displaystyle 5a+52578b=\gcd(52578,5)=1$. Thus we have that $\displaystyle x\equiv x^{5a+52578b}\equiv x^{5a}x^{52578b}\equiv (x^{5})^{a}(x^{52578})^{b}\equiv 1\mod {52579}$ and this completes the proof. The other property that we used about 52579 is that 52578 and 5 are coprime.

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