Science:Math Exam Resources/Courses/MATH312/December 2005/Question 06

MATH312 December 2005

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Question 06
Multiple Choice $\displaystyle 10^{200,000,000,000}$ days from today it will be (a) Sunday (b) Monday (c) Tuesday (d) Wednesday (e) None of the above. Assume that the original exam was written on a Monday.

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 hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.

Hint 1
Note that 7 days from now, we are at the same day we were at today. So it suffices to compute this number modulo 7 then count that many days from today (Monday).

Hint 2
Euler's Formula will help a lot.

Checking a solution serves two purposes: helping you if, after having used all the hints, 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 answer is d = Wednesday Proceeding as in the hints, we need to compute $\displaystyle 10^{200000000000}\equiv 3^{200000000000}\mod {7}$ We use Euler's formula which states that $3^{\phi (7)}=3^{6}\equiv 1\mod {7}$ Next, notice that $\displaystyle 200000000000$ is not divisible by 6 (the sum of the digits is not divisible by 3 so the number is not divisible by 3). However $\displaystyle 200000000004$ is divisible by 6 and thus, $\displaystyle 200000000004-6=199999999998=6s$ is divisible by 6 (here we let s be an integer). Hence $\displaystyle {\begin{aligned}10^{200000000000}&\equiv 3^{200000000000}\mod {7}\\&\equiv 3^{2}\cdot 3^{199999999998}\mod {7}\\&\equiv 2\cdot (3^{6})^{s}\mod {7}\\&\equiv 2\mod {7}\end{aligned}}$ and thus, two days from Monday is a Wednesday.

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