Science:Math Exam Resources/Courses/MATH312/December 2008/Question 03 (b)

MATH312 December 2008

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[hide] Question 03 (b)
Find the least nonnegative integer x that satisfies the following system of linear congruences. $\displaystyle {\begin{aligned}x&\equiv 998\mod {999}\\x&\equiv 999\mod {1000}\\x&\equiv 1000\mod {1001}\end{aligned}}$

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.

[show] Hint 1
Making two clever observations makes this problem much more tractable. First notice that the right hand sides are all congruent to $\displaystyle -1$ modulo the numbers on the far right.

[show] Hint 2
The other observation is that the three moduli on the right are three consecutive pairwise coprime numbers so the Chinese Remainder Theorem will help finish the problem.

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.

[show] Solution
Proceeding as in the hints yields $\displaystyle {\begin{aligned}x&\equiv -1\mod {999}\\x&\equiv -1\mod {1000}\\x&\equiv -1\mod {1001}\end{aligned}}$ As $\displaystyle {\begin{aligned}999&=3^{3}\cdot 37\\1000&=2^{3}\cdot 5^{3}\\1001&=7\cdot 11\cdot 13\end{aligned}}$ we see that all these moduli are coprime. Hence by the Chinese remainder theorem, we have that a unique solution is given by $\displaystyle x\equiv -1\mod {999\cdot 1000\cdot 1001}$ or expanded and made positive $\displaystyle x\equiv 999998999\mod {999999000}$