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

MATH312 December 2008

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Question 01 (d)
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 $\displaystyle 4a\equiv 6\mod {10}$, then $\displaystyle a\equiv 4\mod {10}$.

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
There are two ways to approach this problem. First is trying all 10 possibilities for a and checking that only 4 works. The other involves isolating for 0, factoring and arguing what the possible answers must be.

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 1

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 false We try each combination for a $\displaystyle {\begin{aligned}4(0)&\equiv 0\mod {10}\\4(1)&\equiv 4\mod {10}\\4(2)&\equiv 8\mod {10}\\4(3)&\equiv 2\mod {10}\\4(4)&\equiv 6\mod {10}\\4(5)&\equiv 0\mod {10}\\4(6)&\equiv 4\mod {10}\\4(7)&\equiv 8\mod {10}\\4(8)&\equiv 2\mod {10}\\4(9)&\equiv 6\mod {10}\end{aligned}}$ As we can see, we have two solutions given by $\displaystyle a\equiv 4,9\mod {10}$ and so the statement is false.

Solution 2

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 false. We isolate for 0 giving $\displaystyle 4a-6\equiv 2(2a-3)\equiv 0\mod {10}$ Thus, we have that 10 divides $\displaystyle 2(2a-3)$ and so we are looking for a value of a such that $\displaystyle 5\mid (2a-3)$. We quickly see that $\displaystyle a\equiv 4\mod {5}$ works. This lifts to two solutions modulo 10, namely $\displaystyle a\equiv 4,9\mod {10}$ and this gives a contradiction.

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