Science:Math Exam Resources/Courses/MATH312/December 2008/Question 03 (c)
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Question 03 (c)
Determine all integers x that satisfy the following system of linear congruences.
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!
It's a trap! Be careful about using the Chinese Remainder Theorem here since 6 and 10 are not coprime.
That being said, the same type of idea can be used to get a solution so long as you isolate for 0 and factor at the key step where you cannot invert a non coprime element.
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give us that there is a positive integer s such that
Substituting into the second equation yields
Isolating for 0 gives
Hence, we have that 5 must divide the term in the bracket, that is
Multiplying both sides by 2 gives
Hence for some integer t. Thus,
and hence all solutions are given by