Science:Math Exam Resources/Courses/MATH312/December 2008/Question 03 (a)
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Question 03 (a)
Determine, using no moduli other than 111 in your final answer, all integers x that satisfy the following linear congruence.
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!
First, isolate for 0, factor then argue that one of the terms has to be divisible by 37. Solve that congruence then lift back to 111.
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Proceeding as suggested in the hint, we have
Now, as , we see that we must have
if the above congruence is to hold. Now, isolating gives
We can crunch through the Euclidean algorithm to find the inverse of 7 but here I will use a clever time saving trick. The right hand side is equivalent to
Thus, multiplying by the inverse of 7 on both sides yields
Thus, modulo 111, the solutions are
seen by adding multiples of 37 to 32.