Science:Math Exam Resources/Courses/MATH312/December 2012/Question 05

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MATH312 December 2012

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Question 05
Show that if a and b are relatively prime integers, then $\displaystyle a^{\phi (b)}+b^{\phi (a)}\equiv 1\mod {ab}$ . What is the multiplicative inverse of $\displaystyle 5^{8}$ modulo 16?

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
For the first part, use Euler's Theorem. For the second part, simply evaluate it.

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

Euler's Theorem states that ${\begin{aligned}a^{\phi (b)}&\equiv 1\mod {b}\\b^{\phi (a)}&\equiv 1\mod {a}\end{aligned}}$ Translating the above, there exist integers s and t such that ${\begin{aligned}a^{\phi (b)}-1&=bs\\b^{\phi (a)}-1&=at\end{aligned}}$ Taking the products of the left hand sides of the equation and the right hand sides of the equation yields (below we have flipped the sides of the equations) ${\begin{aligned}abst&=(a^{\phi (b)}-1)(b^{\phi (a)}-1)=a^{\phi (b)}b^{\phi (a)}-a^{\phi (b)}-b^{\phi (a)}+1\end{aligned}}$ As $\displaystyle \phi (a)$ and $\displaystyle \phi (b)$ are at least one, we have that ${\begin{aligned}0&\equiv -a^{\phi (b)}-b^{\phi (a)}+1\mod {ab}\end{aligned}}$ and hence we get the required result. For the last part, we have $\displaystyle 5^{8}\equiv (5^{2})^{4}\equiv 25^{4}\equiv 9^{4}\equiv (9^{2})^{2}\equiv 81^{2}\equiv 1^{2}\equiv 1\mod {16}$ (or easier, notice that $\displaystyle \phi (16)=8$ so Euler's theorem gives the same result) Thus as the number is 1, its inverse it itself.

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