[Undergraduate Discrete Math] Modular Math Proof
I am reviewing math that I have done over the past couple of years. Here is a link to a image of the problem and proof that I have. Please give me input, suggestions etc.
Posted May 25, 2014, 11:30 PM UTC
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Typically the statements "a=b (mod n)", "n divides a-b", and "exists q such that a-b=qn", are equivalent by definition. What definition are you using for a=b (mod n)?
This from the particular chapter in the text that I am using (Gallian Contempary Abstract Algebra 6th edition):
When a=qn+r where q is the quotient and r is the remainder upon dividing a by n, we write a mod n = r or a = r mod n.
The proof should be enough. Depending on how pedantic you need to be, for the first part, you can also note why (a-b) (mod n) = 0 implies that a (mod n) - b (mod n) = 0 (by modular arithmetic).
For the second part, you can also note that q1, q2 and r are unique, given m and non-zero n, by the division theorem.
If not, that should be fine.
Thanks for taking the time to look it over.