Work in progress: this question page is incomplete, there might be mistakes in the material you are seeing here.
Question 07 (ii)
Let such that and let .
(ii) For each odd integer x, prove that there exists a unique and a unique such that
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!
Part (i) shows us that the order of a modulo is equal to . Thus each of
are distinct odd numbers and thus each of the
are also distinct odd numbers. Hence it suffices to show that
for any and finish off the proof with a counting argument.
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.
Repeating the hint, part (i) shows us that the order of a modulo is equal to . Thus each of
are distinct odd numbers between 0 and for and thus each of the
are also distinct odd numbers between 0 and . As there are total odd numbers between 0 and , we know that provided all of the above numbers are distinct, we have found total odd numbers and hence we can form a bijection. Hence it suffices to show that
for any . This shows the claim for all odd numbers since all odd numbers are equivalent modulo to an odd number between 0 and .
Assume towards a contradiction that we found two such numbers:
Without loss of generality, suppose that . Then,
Once again, modulo 4 arguments (recalling that ) show that the left hand side above is congruent to 1 modulo 4 and the right hand side is still congruent to -1 modulo 4. This is a contradiction. Thus any odd number can be expressed uniquely of the form
with and as required.