Science:Math Exam Resources/Courses/MATH312/December 2012/Question 01 (a)

MATH312 December 2012

  •  Q1 (a)  •  Q1 (b)  •  Q1 (c)  •  Q1 (d)  •  Q1 (e)  •  Q1 (f)  •  Q1 (g)  •  Q2 (a)  •  Q2 (b)  •  Q2 (c)  •  Q2 (d)  •  Q3  •  Q4  •  Q5  •  

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Question 01 (a)
Short answer questions: Each question carries 6 marks, your answers should quote the results being used and show your work. Find two integers congruent to 3 modulo 5 and 4 modulo 7.

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
Use techniques related to the Chinese Remainder Theorem.

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. We seek to solve the system of congruences given by ${\begin{aligned}x&\equiv 3\mod {5}\\x&\equiv 4\mod {7}\end{aligned}}$ The first equality says that $\displaystyle x=3+5s$. Plugging into the second yields $\displaystyle 3+5s\equiv 4\mod {7}$. Simplifying yields $\displaystyle 5s\equiv 1\mod {7}$. We can find the inverse of 5 by using the Euclidean algorithm however since we are modulo 7, there is only 6 possible candidates so it is faster to just try them all. Since $\displaystyle 5\cdot 3\equiv 15\equiv 1\mod {7}$, we see that $\displaystyle s\equiv 3\mod {7}$. This is the same as $\displaystyle s=3+7t$. Back substituting gives $\displaystyle x=3+5(3+7t)=18+35t$. So two possible answers are given by 18 and 53.

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